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LIBRARY OF CONGRESS, 



ChapQLQMCopyright No. 
Shelfi 



TiiD STATES OF AMERICA. 



EXERCISES 



PHYSICAL MEASUREMENT 



LOUIS W/AUSTIN, Ph.D., (strassbukg) 

/ AND 
CHARLES Br THWIKG, Ph.D. (bonx) 

Instructors us Physics i> t the University of Wisconsin 




Boston 

ALLYN AND BACON 



1896 



I 



Q 






c,* 1 
^ 



Copyright, 1895, by 
Louis \V. Austin and Charles B. Thwing. 



Typography hy C. J. Peters & Son, 
Boston. 



PREFACE. 



This little volume does not profess to be a complete 
handbook of physical practice. That place is well filled 
by the last German edition of Kohlrausch, by Stewart & 
Gee, and others, all of which, it is presumed, are found 
upon the reference shelves of every physical laboratory. 
It has been our aim, rather, to put in compact and con- 
venient form such directions for work and such data 
as will be required by the student in his first year in the 
physical laboratory. 

This course presupposes such a knowledge of the prin- 
ciples of physics as can be gained from a course of general 
lectures, supplemented by a good textbook. It has been 
found practicable to carry on the course parallel with the 
lectures by occupying the first month of the year with 
simple measurements, so as to allow the remainder of the 
work to be taken up after the principles involved have 
been traversed in the lectures. 

The exercises in Part I. are essentially those included 
in the Practicum of the best German universities, except 
that a few of the more difficult and those requiring appa- 
ratus too expensive to be duplicated are reserved for the 



iv mil: FACE. 

second year's work. They are exclusively quantitative; 
our own experience having confirmed us in the opinion 
that illustrative experiments are best confined to the lec- 
ture-room, while quantitative work, on the part of the 
student himself, is admirably adapted both to arouse and 
maintain his interest, and to cultivate habits of accuracy 
and thoroughness. 

These experiments have heretofore been performed 
by our own students, working from manifolded copies. 
While putting them in more permanent form, we have 
thought best to add such suggestions with regard to com- 
putations and some of the more important physical manip- 
ulations as would make unnecessary the purchase by the 
student of a second laboratory manual. This information 
has, for convenience, been put at the end of the exer- 
cises, and comprises Part II. 

The apparatus required for the exercises is for the 
most part inexpensive, or such as is found, or can be 
readily made, in any well equipped laboratory. 

Part III. contains in tabular form such data as will be 
needed by the student in making his computations and 
verifying his results. The logarithmic and trigonometric 
tahles are preceded by brief directions for their use. 

Except in certain details, no special claim is made to 
originality in the matter presented in the book. Among 
the works consulted in its preparation are the manuals of 
Kohlrauseh. Stewart & (ice, Wiedemann & Ebert, Sabine, 
and Nichols. In the preparation of the tables we have 
followed largely Kohlrauseh, the Smithsonian Tables, and 



P RE FACE. V 

the new edition of Landolt and Bornstein. In the mathe- 
matical tables the arrangement used in Bottomley's Four 
Figure Mathematical Tables and Nichols's Laboratory Man- 
ual has been followed. To all of these authors we wish 
here to express our obligations. 

Our thanks are especially due to Prof. Benjamin W. 
Snow for many valuable suggestions in the preparation of 
Part II., and to Professor Snow and Prof. Chas. S. Slichter 
for assistance in reviewing the manuscript. 

We shall esteem it a special favor on the part of 
physicists if they will call our attention to any errors or 
to any obscurity of expression which may come to their 
notice. 

LOUIS W. AUSTIN. 
CHARLES B. THWIXG. 
Madison, Wisconsin, 

Sejjtembt?*, 1895. 



CONTEXTS. 



PAGE 

Introduction 1-6 

PART I. PHYSICAL MEASUREMENT. 

Extension — 

The Vernier 7 

The Vernier Gauge 8 

The Barometer 9 

The Micrometer 11 

The Spherometer 12 

The Filar Micrometer 14 

The Dividing Engine 15 

The Cathetometer * 16 

Mass — 

The Balance 18 

Weighing 22 

Batio of the Arms of a Balance 28 

Weight in Vacuo 24 

Density — 

Density with the Specific Gravity Bottle 26 

Density by Immersion 27 

Density of a Vapor 28 

Gravity — 

Time of Oscillation of a Pendulum 31 

Acceleration of Gravity with Simple Pendulum 31 

Elasticity — 

Use of the Telescope and Scale 34 

Young's Modulus by Stretching 34 

vii 



vni CONTENT*. 

Elasticity (Continued)— page 

Young's Modulus by Bending 38 

Modulus of Torsional Elasticity 39 

Moments of Inertia 40 

Time of Vibration by Middle Elongations 41 

Capilla kity — 

Capillary Constant in Tubes 44 

Surface Tension by Size of Drop 46 

Viscosity — 

Coefficient of Molecular Friction by Flow 46 

Heat — 

Calibration of Tliermometers 48 

Coefficient of Expansion 51 

Specific Heat 54 

Latent Heat 56 

Mechanical Equivalent of Heat 57 

Hygromktjry — 

Vapor Pressure and Humidity 60 

Dew Point by DanielPs Hygrometer 61 

Introduction to Electkictpy and Magnetism — 

Use of Mirror, Telescope, and Scale 62 

Batteries 63 

Galvanometers 67 

Magnetism — 

Angle of Inclination 72 

Horizontal Intensity of Earth's Field 73 

Kl K< TKICTTY — 

Ptesistance by Substitution 77 

Wheatstone' s Bridge 78 

Specific Hcsistance 84 

Temperature Coefficient of Resistance 8C 

Resistance of Electrolytes 87 

I Resistance by the Differential Galvanometer 89 

Calibration of Galvanometers 91 

Electromotive Force 95 

The Quadrant Electrometer 96 



coy TEXTS. lx 

Sound — page 

Telocity of Sound by Kundt's Method 99 

Velocity of Sound with Konig's Apparatus 100 

Pitch of a Fork with the Siren 101 

Light — 

Photometry 102 

Lenses 104 

The Spectrometer 10S 

Spectrum Analysis 112 

The Diffraction Spectrum 114 

Polarization 117 

Saccharimetry IIS 

PART H. (A). DISCUSSION OF RESULTS. 

Errors 121 

Calculations with Small Quantities 123 

Method of Least Squares 124 

PART H. (B). MANIPULATIONS. 

Manipulations with Glass 132 

Quartz Fibres 13S 

Soldering 140 

Cleaning Mercury 141 

Cements and Tarnishes 142 

Construction of Standard Cell 143 

Construction of Water Batteries 144 

PART m. TABLES. 

Density of Solids 14S 

Density of Liquids 150 

Density of Gases 151 

Corrections for Loss of Weight in Air 153 

Hygrometry . .' ' 154 

Barometer Corrections 155 

Thermometer Corrections 150 

Melting-Points and Boiling-Points 157 

Critical Temperatures 15S 

Latent Heats 15S 

Specific Heats 159 

Heats of Combustion 160 



x COXTENTS. 

PAOS 

Bcients of Exx>ansion 101 

Thermal Conductivity 101 

Freezing Mixtures 102 

Capillary Constants 162 

Coefficients of Friction 103 

Elasticity of Solids 103 

Velocity of Sound 104 

Pitch and Vibration Number of Xotes 164 

Corrections for Oscillations in Large Arc 105 

Positions of Lines in Spectrum 166 

Wave-lengths of Fraunhofer's Lines 167 

• r of Newton's Kings 168 

Indices of Refraction . . . 160 

Electromotive Force of Batteries 169 

Electrical Resistances 170 

Temperature Coefficients of Electrical Conductivity 171 

Specific Inductive Capacities 171 

Magnetic Constants for the United States 172 

Numerical Constants 175 

Physical Constants 176 

Conversion of English to Metric Units 177 

Conversion of Metric to English Units 178 

Dimensions of Physical Quantities 180 

Directions for Use of Logarithm Tables 181 

^a/ithm and Trigonometric Tables 184 

Index o ....... . 195 



INTRODUCTION. 



The object of the experiments described in the follow- 
ing course is to teach the student the use of the instru- 
ments and the methods of observation employed in physical 
measurements, and to introduce him to such methods of 
work as will fit him at a later period to undertake original 
investigation. 

Before beginning an experiment the directions should 
be read carefully, and the apparatus examined until its use 
is fully understood. 

The most essential part of a physical measurement is 
the making of one or more observations, upon the accuracy 
of which depends the value of the results obtained. The 
following are some of the most common errors of observa- 
tion, with some suggestions concerning methods of avoid- 
ing them. (1} Error of the zero: Let the first 

. v Errors. 

reading taken be the zero. If this is found to 
be other than the zero of the scale, the amount of vari- 
ation should be determined, and the correction applied to 
the observed readings. (2) Error of parallax: All obser- 
vations should be made with the eye in a line normal 
to the surface at the point observed. In reading a ther- 
mometer, for example, if the eye be above or below the 
normal, the position of the mercury with reference to the 
scale will appear to be lower or higher than is actually 

1 



2 IXTKODUCTION. 

the case. Since the object to be measured is seldom 
in actual contact with the scale, nearly all naked eye 
observations are liable to this error. (3) Errors of preju- 
dice: Let every observation made be wholly uninfluenced 
by any previous observation, either of your own or of 
others. It is, when possible, well to cover the scale when 
setting an instrument. (4) The effect upon the result of 
accidental errors may to a great extent be eliminated 
by taking the average of a large number of observations 
of the same quantity. 

In general no observations should be rejected on 
account of any suspected error, unless the cause of the 
Rejection of error is evident. It is allowable, when there 
observations. are some discordant observations in a series, to 
compute two results, one in which all the observations are 
included, and another with the suspected observations 
thrown out. Every observation taken in the series, how- 
ever, should always be given; and, if any are not used, 
the reasons for excluding them should be explained. 

Often, when the final result sought is not measured 
directly, but is derived from the observations by means 
Relative of a formula, errors in some of the observations 
importance w jjj ] mve a muc h greater effect on the result 
observations, than in others. Such observations must be 
taken with proportionately greater care and accuracy. 
This question should always be considered before an 
experiment of this sort is begun. For example, in Exer- 
cise 20, an error in the measurement of L or h will affect 
the result much more than an error in measuring I or b. 
The measurement of these quantities must therefore be 
taken with especial pains. 

On most instruments the smallest divisions of the 
scale aire large enough so that tenths of these divisions 



IX TB OD UCTIOX. 3 

may be estimated with the eye. This estimation, though 
not always perfectly accurate, will give more 
correct results than as though the fraction of S T^nth S n 
were neglected. 

A complete and orderly record of all work done should 
be preserved in the form of a permanent notebook.* The 
following is the logical order of recording the Record of 
work : — work. 

(1) The object and method of the experiment. 

(2) A working description of the apparatus used. 

(3) The record of every observation made, including 
all incidental facts, such as temperature of the room, height 
of the barometer, or the like, which might in any way 
affect the results observed. The actual observations are 
here meant. Thus, if the student is finding the difference 
in time between two events, he should note in his book the 
actual time of the first and the second, not the difference 
directly, which is a derived result found by subtracting 
the time of the first from that of the second. All related 
observations should be recorded in tabular form, and a 
curve platted where graphic representation is desirable. 
The drawings should be numbered consecutively and the 
parts lettered, so that they may be referred to in the 
description. 

(4) A discussion of the results obtained. For all 
multiplications and divisions not easily performed in the 
head, logarithms should be used. Since the computa- 
degree of accuracy possible to be attained in tions - 
any experiment is limited by the instruments and methods 
used, it is useless to carry the calculations to a larger 
number of places of figures than is warranted by the 

* Tlie original observations should not be trusted to loose sheets of 
paper, but be put down at once in a temporary notebook. 



4 INTRODUCTION. 

refinement of the observations. All results obtained by 
calculation maybe given to three places of figures, unless 
otherwise indicated in the exercises. The number of 
places of figures to be given in the result will be in every 
case one place more than the limit of required accuracy. 
By "number of places'' is meant the number of signifi- 
cant figures. Thus the results 1900. and 0.0022 are both 
given to two places of figures. 

In order that all the figures given in the result may be 
free from errors of calculation, the computations should 
be carried one place farther than the number of places 
to be given in the result. If the result is to be given to 
four places of figures, for example, all figures beyond the 
fifth place should be dropped throughout the computa- 
tions. When the first figure dropped is five or more, one 
must be added to the last figure retained. 

The greatest care should be taken to avoid numerical 
errors, either in copying or in calculation. All computa- 
tions should be gone over at least twice, even if the result 
obtained appears to be the one desired. 

The significance of a series of related results may 
often be brought out more fully by platting them in 
Graphic Rep- the form of a curve. For example, suppose 
resentation. ±] m ^ ±\ w sensibility of a balance for varying 
loads Avas found to be as follows: — 

LOAD. SENSIBILITY. 

o g. 1.G40 

10 " 1.800 

20 • 1.858 

30 • 1.798 

lo •• 1.652 

In platting the curve ("see Fig. 1), it is desirable to 
choose our units of length along each axis of such di- 



LOAD. 


SENSIBILITY. 


50 g. 


1.512 


60 " 


1.390 


80 " (1.28) 


1.188 


00 " 


1.022 



INTB OD UCTION. b 

mensions, that the curve shall as nearly as possible occupy 
the whole of the sheet of paper on which it is traced. In 
our example we chose 2 mm. as the length upon the axis 
of X corresponding to a weight of one gram, and 2 mm. as 
the length on the axis of Y corresponding to 0.01 of a 







^■^^•^ 


S N ~X- 






J v^ 


y js jz 


Z ^ 


17 J V- 


17 t \ 


1 ^Sv- T + ^ 


( K 




4 r> \ 




\ 


^ 


V 






\ 


^ 


s. 


\ 


7.2 " "ll 


^. 


>. 


> 


- \ 






s 


> 






V 


^< 


^ 


hi s 


^ 


XT 


S s, 


~^~ -D 




' 








0.9 ... 1 1 J_ 



40 50 

Fig. 1. 



SO 



scale division. Since our lowest value for the sensibility 
is above 1, and our highest below 1.9, we should so lay off 
our paper that these values lie respectively near the 
bottom and the top of the sheet. 

In cases where the curve is regular, that is, where 
the quantities vary according to a definite law, the curve 



b INTRODUCTION. 

will enable as to determine, with considerable accuracy, 

values Intermediate between those obtained by direct 

observation. Thus we see that the point on 

Graphic J - 

Interpola- tllC Clll'VC ('( )1TCS] )011(lil)0' to a loatl of 35 g. IS 

1.724, and for all practical purposes this may 
be taken as the correct value. 

Another use of the curve is to call attention to inaccu- 
rate observations. Thus, if a point lies at some distance 
out of the general direction of the curve, it is 

The Curve as ° 

a check on likely that the observation is in error. In the 
example the sensibility for a load of 80 grams 
was in error, due to a mistake in division. The corrected 
result falls in its proper place on the curve. 



PART I. 

PHYSICAL MEASUREMENT. 



MEASUREMENT OF EXTENSION. 



THE VERNIER. 

The vernier provides a method of estimating fractions 
of the smallest divisions of a scale with much greater 
accuracy than is possible with the unaided eye. It con- 
sists of a movable scale which slides upon the fixed scale. 
In the common form of straight vernier the movable or 
vernier scale is divided into ten divisions, each of which 
is 0.9 of a division of the fixed scale. (See Fig. 2.) 
When the quantitj^ measured is an exact number of divis- 
ions of the fixed scale, the first and last divisions of the 
vernier, and those onlj^, will coincide with divisions of the 
fixed scale. If the quantity measured is not an exact 
number of scale divisions, the fractional part of a division 
may be determined as follows : — 

Observe which division of the vernier coincides with a 
division of the fixed scale. The number of this division 
of the vernier will be the number of tenths of a scale 
division to be added to the reading of the fixed scale to 
give the exact measurement. For example, the exact 
length of the object measured by the vernier gauge 
shown in Fig. 2 is 12.5 mm. This is true, since each 

7 



8 



/>// )'si( \ 1 L M E. isr E /•;.)/ ENT, 



division of the vernier being { \ } shorter than the divisions 
of the scale, division 4 of the vernier must be 0.1 mm. to 
the right of division 1(> of the scale. In like manner, 3 
of the vernier is o.*2 mm. to the right of 15 of the scale, 
division '2 of the vernier 0.8 mm. to the right of 11 of the 
scale, division 1 of the vernier 0.4 mm. to the right of 13 
of the scale, and of the vernier, which is the point 
measured, is 0.5 mm. to the right of 12 of the scale. 

The vernier gauge, Fig. 2, consists of a graduated 
steel scale having a fixed jaw at one end and a movable 
jaw arranged to slide slong the scale always parallel to 
the fixed one. When the jaws are closed, the zero of the 




Mill llJII 


MM IN MM 

11 ' l2 



T 



MI|||lll|IMI|llll|MM|MM|llll|IIIMIMMIIII|l]||| 



II I I I M II 






I I I I 


I I I I l 


I I I I 


1 


Jin 


I 


". 










Fig. 2. 

vernier coincides with the zero of the scale. A set screw, 
S, attached to the movable jaw, enables it to be set in any 
desired position. 

Exercise 1. — To measure the length and diameter of a 

cylinder. 

First close the jaws, and note the correction, if any, 

for zero. To measure the diameter, place the cylinder 

between the jaws, close them with a gentle pressure, set 



MEASUBEMENT OF EXTENSION. 9 

the screw, and take the reading. Next make a second 
measurement along a diameter at right angles to the first. 
Make five pairs of such measurements at about equal dis- 
tances along the length of the cylinder. Then average 
results, and apply the correction for zero. 

Example : — 

MEASUREMENTS. DIAMETERS. 

1 12.4 

2 12.5 

3 12.4 

4 12.5 

5 12.3 

6 12.4 

7 12.4 

8 12.4 

9 12.3 

10 12.4 

Average 12.36 12.44 12.4 

Error of zero 2 

12.2 



THE BAROMETER. 

The most common form of mercurial barometer is that 
of Fortin. (See Fig. 3.) By the height of the barome- 
ter is meant the difference in height of the two mercury 
surfaces, S and S' . The zero of the scale is the tip of the 
ivory point, p, which projects into the cistern. Hence, that 
the scale may measure the height of the column, the sur- 
face, #, must be adjusted by means of the adjusting-screw, 
a, till the ivory point just touches the surface of the mer- 
cury. The reading is taken on the vernier, V, which is 
moved by means of the milled head, m, till the light reflected 
from the white background just disappears at the middle 
point of the surface, S'. To prevent parallax the vernier 
is graduated on a movable tube. The eye is in the proper 



Ill 



/'// TSH ■. I /. MEASUREMENT. 



m 




Fig. 3. 



position for reading when the front lower edge 
of this tube appears to coincide with the back 
lower edge. The vernier differs from the form 
already described, only in that the ten divisions 
are subdivided into halves. It is possible to 
read to 0.05 mm., since 20 of these half divis- 
ions correspond to 19 mm. on the scale. For 
r _ example, the reading of the vernier in 

Fig. 3 is 762.15 mm. 

Exercise 2. — To take a series of 
readings of the barometer. 

Adjust the height of the mercury 
in the cistern. Once adjusting is suf- 
ficient during the time of the exercise. 
Take readings every ten minutes for 
an hour. Each reading should be the 
77 mean of three independent settings of 
the vernier. Record the time (date, 
hour, and minute) of making the ob- 
servations. Tabulate the results, and 
plat them in a curve, using time as 
abcissas and the, height of the baro- 
meter as ordinates. 

Corrections. — (a) For temperature. The 
true height of the barometer is its height at 0° C. Since 
mercury expands 0.00018 of its volume for each degree C, 
the true reading at zero would be : b = h — 0.00018 'h% 
where h = the observed height and t = the temperature. 
A further correction is made necessary by the expansion 
of the brass scale, if it be graduated for 0° C. Brass ex- 
pands 0.000010 of its length for each degree. With this 
correction we have 

b = h — (0.00018 - 0.000019) -h't 
(See Table 13.) 







\m 


— 


» 




** 


- 78 


- 


— 


*o , 


" 77 




— 


o 






~ 76 



MEASUREMENT OF EXTENSION. 



11 



(6) For capillarity. A correction is sometimes added, especially if 
the barometer tube is of small diameter, for capillary depression. The 
amount of this for different diameters is given in Table 13. 



THE MICROMETER. 

A still more exact instrument than the vernier for 
measuring small dimensions is the micrometer, the prin- 
cipal forms of which are the micrometer gauge, the 
spherometer, the filar micrometer, and the dividing 
engine. 

The micrometer gauge (Fig. 4) consists of a bent 
arm, one end of which is threaded to receive a screw, the 




Fig. 4. 



other end containing an adjustable stop, S. The head of 
the screw is divided into equal parts for reading fractions 
of a turn. The longitudinal scale on the frame is divided 
to mms. If the distance between the threads of the screw 
be ^ mm., the screw will advance 1 mm. for every two 
turns. If the head be divided into fifty parts, turning 
through one of these divisions will advance it 0.01 mm. 
If the distance between the threads be 1 mm., the head 
will usually be divided into one hundred parts, and will 
read to 0.01 mm. as before. 



L2 



PII FS K \ 1 L M E. 1 S UREMENT. 



Exercise 3. — To measure the diameter of a steel rod. 

First bring the screw and stop in contact with a gentle 

pressure. .V uniform pressure may always he obtained by 

grasping the head of the screw loosely in the 

lingers, and turning till the fingers slip. If the 

zero is not found to be correct, it should be 

corrected by adjust- 
ing the stop, or 
correction must be 
made for the error 
of the zero in re- 
cording the meas- 
urements. Find the 
average of a series of ten 
readings at equal distances 




along the length of the rod. 

The spherometer (Fig. 5) 
consists of a tripod, the legs 
of which form an equilat- 
eral triangle, with the screw 
in its centre. The head of 
the screw is enlarged to form 
a disk, which, if the pitch of 
the screw be J mm., is divided 
into 500 parts. The instru- 
ment will then read to .001 
nun. The number of turns of the screw is read from the 
vertical scale, S. It is of the greatest importance to deter- 
mine the instant of contact with precision. For this pur- 
pose the instrument is often provided with a pointer, _P, 
which rises at the instant of contact. When this device 
is not provided, the instrument will, when the screw has 



Fig. 5. 



MEASUREMENT OF EXTENSION. 13 

been turned a little too far, begin to rock and perhaps 
revolve with the screw. The screw must then be turned 
back till this motion ceases, when all of the four feet will 
be in contact with the surface. 

Exercise 4. — Measurement of the thickness of a small glass 
plate with the spherometer. 

First place the instrument upon a large piece of plate 
glass, and determine the zero * readings from a series of 
five observations taken on different parts of the plate. 
Next raise the screw, place the glass plate to be measured 
under it, and take a series of readings on the plate at 
about equal distances apart. The average of these read- 
ings subtracted from the average zero reading will give 
the average thickness of the plate. 

Exercise 5. — Radius of curvature of a lens -with the sphe- 
rometer. 

Without special appliances, only lenses of a diameter 
large enough for the spherometer to stand upon can be 
measured. Raise the screw till it is not in contact when 
the spherometer is placed upon the lens, then take 5 read- 
ings of the position of the contact screw for different points 
upon the lens. The average of these readings subtracted 
from the zero reading will give the height, h (see Fig. 6), 
of the point of contact of the screw, p, above the plane of 
the other 3 points. If d be the distance from the point 
of the screw to each of the other 3 points, when they are 
all in the same plane, then by geometry the radius — 

h 2 + d 2 



B= -Vd' 2 + (B - h) 2 = 



2h 



* Since the instrument is used for measuring both concave and convex 
spherical surfaces, the " zero reading " will lie near the middle of the scale. 



14 



/■// ran al measurement. 




Since the distance between the fixed, legs, I, is larger than 
d, it can be measured more exactly. By geometry, — 



d = 



V3 



Then in terms of Z, — 



To obtain 7, place the instrument upon a smooth sheet of 
paper, and press it down slightly to get the impressions 
of the -3 points. The distance I may then be measured by 
means of the vernier gauge or dividers and scale. Since 
the legs may not form a perfect equilateral triangle, the 
average of the three measurements is to be taken. For a 
concave lens, h is a negative quantity; and the formula 
remains the same, with the exception that R is negative. 

(iive result to four places of figures. 

The filar micrometer consists of a micrometer screw 
which is arranged to move a fine thread across the field of 
view of a microscope or telescope. The number of turns 
is usually indicated by a scale at one side of the field. 



MEASUREMENT OF EXTENSION. 15 

Exercise 6. — - To determine the constant of a filar microm- 
eter, and to measure the diameter of a small object. 

When the instrument has a constant focus, as in the 
microscope, proceed as follows : Focus the instrument on 
a standard scale, and determine from a series of trials the 
number of turns of the screw required to move the thread 
over 1 mm. of the scale. The reciprocal of this number 
will be the apparent distance passed over by the thread 
for each turn of the screw. This number expressed in 
decimals of a mm. is known as the constant of the filar 
micrometer. To measure the diameter of a small object. 
turn the screw till the thread exactly coincides with the 
right-hand side of the object. Note the reading, and turn 
the screw till the thread coincides with the left-hand side. 
When possible, the average of a number of readings, with 
the object in different positions, should be taken. 

Caution. To avoid any error from lost motion due to the 
serew fitting too loosely in its nut. the thread should always be 
brought to the jyosition for measurement by a forward motion 
of the serew. The same precaution should be observed in any 

instrument where lost motion is jiossible. 

The dividing engine consists essentially of a sliding 
table, upon which the object to be measured rests. This 
table is propelled by a micrometer screw, and above it is a 
stationary microscope provided with cross-hairs in the eye- 
piece for accurately sighting upon a point of the object to 
be measured. 

Exercise 7. — To calibrate a scale -with the dividing engine. 

Clamp the scale firmly to the sliding table parallel to 
its direction of motion, and turn the screw till the zero 
point of the scale coincides with the intersection of the 



16 



PHYSICAL MEASUREMENT. 



cross-hairs; note the reading, and turn the screw till the 
first division of the scale coincides with the cross-hairs. 
Take similar readings on the remaining scale divisions. 

Knowing the constant of the 
screw, the actual values of the 
scale divisions are then known, 
and a table of corrections for the 
scale may be formed. 

If the dividing engine is to be 
used for ruling scales, a cutting 
instrument will be substituted for 
the microscope. In this case the 
successive distances turned by the 
screw are regulated by a ratchet 
wheel, which may be so divided 
as to give divisions of any de- 
sired length. For a more com- 
. plete description of the dividing 
engine the student may consult 
Steward & Gee. 

THE CATHETOMETER. 

The cathetometer (Fig. 7) is 
an instrument for measuring ver- 
tical distances less than a metre. 
It consists of a heavy vertical bar, 
_B, upon which is ruled a scale. 
This bar is supported upon a 
tripod provided with levelling 
screws, and is capable of rotation about a vertical axis. 
A telescope, T, to which is attached a spirit level, is sup- 
ported upon a carriage, (7, which slides upon the bar, J?, 
and may be clamped at any point upon it by means of a 
screw, I). 




Fig. 7. 



MEASUREMENT OF EXTENSION. 17 

The following adjustments are necessary : — 

1. The intersection of the cross-hairs in the eyepiece 
must be in the line of collimation of the telescope. To make 
this adjustment, set the telescope upon a dot so that the 
point of intersection of the cross-hairs appears to coincide 
with it, and rotate the telescope about its own axis. If 
the point of intersection moves from the dot, it must be 
brought half-way back by means of the screws supporting 
the ring which carries the cross-hairs, and the remaining 
half-way by moving the telescope vertically by means of 
screw E. Repeat this operation until the point of inter- 
section remains upon the dot while the telescope is 
rotated. 

2. The level must be parallel to the axis of the telescope. 
First adjust the telescope screw, #, which controls the 
angle between the telescope and the bar until the bubble 
is in the middle of the level tube, then reverse the tele- 
scope, turning it end for end in its Y s 5 an( i ^ ** ^ s 
not found to be level in its new position, bring it half- 
way back by means of the adjusting screws of the level, 
and half-way by means of the screw S. Repeat until the 
bubble remains in the centre in both positions. In observ- 
ing the level note the position of both ends of the bubble. 

3. The telescope must be at right angles to the bar, and 
the bar must be vertical. Unclamp B, and turn it until the 
telescope is in a line parallel to the line joining two of the 
levelling screws, for instance, A 1 and A 2 ; clamp it and 
level the telescope; then turn B through 180°, and bring 
the telescope back to level partly by adjusting A 1 and 
A 2 , and partly by the screw S. Now turn B through 
90°, and level again, using A s . Repeat until the bubble 
remains in the middle of the tube during a complete 
revolution. 



18 PHYSICAL MEASUREMENT. 

/../ rdse 8. — To measure the difference in height of two points. 
Loosen the clamp Z), support the telescope carriage 
with the hand, and raise or lower it till it is nearly on a 
level with one of the points to be observed. Set the 
clamp JK rotate the bar till the vertical cross-hair coin- 
cides with the point, and clamp the bar. The telescope 
may now be adjusted by the slow-motion screw E till the 
horizontal cross-hair coincides with the point, and the read- 
ing taken by the scale and vernier. By exactly the same 
proceeding the height of a second point may be deter- 
mined. The difference between the two readings will be 
the vertical distance between the two points. 



MEASUREMENT OF MASS. 



THE BALANCE. 

The balance is an instrument for comparing masses. 
It consists essentially of a beam which is capable of rota- 
tion at its centre upon a knife-edge which rests upon an 
agate plate fixed at the top of a supporting central column. 
At the ends of the beam, at equal distances from the centre, 
are supported, also by means of knife-edges and agate 
plates, two pans which carry the masses to be compared. 
By means of the arrestment, a, operated by the milled 
head, 37, the weight of the beam is taken from the knife- 
edges whenever the balance is not in use. In the best 
balances there is also provided an arrestment for the pans, 
which is operated in connection with the arrestment for 
the beam, but is sometimes operated separately by a push 
button. />. placed at the left of the milled head. The 
pointer, />. connected with the beam, swings over a grad- 



MEASUREMENT OF MASS. 



19 



uatea scaie at the bottom of the pillar, P. To avoid 
negative readings the centre of the scale is counted as 10, 
the zero being at the left. On account of the incon- 
venience of handling very small weights, the beam is gracl- 




Fig. 8. 



uated, and a wire rider provided, which may be dropped 
at any point on the beam by means of the lever, /, without 
opening the case. If the twelfth division of the- arm 
comes directly over the point of support of the pan, it is 
evident that the rider must weigh twelve mg., for in this 
position it has the same moment as when placed in the 



20 PHYSICA L MEASUREMENT. 

pan. Its effective moment at any numbered division will 
then be the number of nigs, indicated at that division. 

The Large weights in a set are marked in grams. The 
small weights, it' marked in decimals, are fractions of a 
gram, if marked in whole numbers, milligrams. The set 
is so made up that it is possible by proper combinations 
of the weights to make any desired weighing. The small 
weights of a set should consist of: — 



A t>" 



One 500 mg. Two 100 mg. One 20 mg. 

One 200 " One 50 " Two 10 " 

If no rider is used, there should also be, — 

One 5 nig. Two 2 mg. One 1 mg. 

Since the mixing of weights from the different sets is a 
cause of great inconvenience in the laboratory, the weights 
of one set should never be taken to use with those of 
another. 

No greater load should ever be placed on a balance 
than can be weighed with the weights in the set which 
goes with it. 

GENERAL RULES FOR THE USE OP THE BALANCE. 

1. See that the rider is free from the beam, so as not to 

touch it during the swing. 

2. Determine the zero point anew before each weighing. 

The zero should not vary more than one scale 
division from the centre. 

3. Do not stop the swings of the beam with a jerk. This 

is best avoided by raising the arrestment just as 
the pointer is crossing the centre of the scale. 

4. Always handle the weights with the pincers, not with 

the fingers. 

5. Never completely lower the arrestment when the 

equilibrium is not perfect enough to prevent the 



MEASUREMENT OF MASS. 21 

centre of swing of the pointer being more than 
five divisions from the centre of the scale. A very 
slight lowering will be sufficient to determine 
this. 

6. Raise the arrestment whenever weights are to be 

added or removed. 

7. Place the object to be weighed in the centre of the 

left-hand pan, the large weights in the centre of 
the other pan, adding the small weights in the 
order of their size. 

8. In observing the swings, the case mnst be closed to 

prevent air-cnrrents, and the observer shonld take 
a central position to prevent parallax. The 
swings should not be more than five divisions in 
length. 

9. Substances liable to injure the pans must be weighed 
in a watch-glass or glass beaker, volatile sub- 
stances in a stoppered bottle. 

10. When the weighing is finished, raise the arrestment, 
lift the rider, put the weights in their respective 
places in the box, and close the case. 

The above rules are very important, and should be 
carefully remembered. 

Note. — Since the weight of a body, when measured with the lever 
balance, is proportional to its mass, the word weight will be used in its 
commonly accepted sense of mass, except when otherwise indicated. 

Exercise 9. — To determine the zero of the balance. 

As it is a waste of time to wait for the pointer to come 
to rest, the position of equilibrium may be determined by 
observing the extreme position of the pointer for three 
successive swings.* Since the length of the swing is grad- 
ually growing less, the average of the first and third of 
these swings will be as far to one side of the zero as the 



22 PHYSICAL MEASUREMENT. 

second is to the other; hence the average of the mean of the 
light-hand readings with the mean of the left-hand read- 
ings will be the zero point. A somewhat more exact valne 
may be obtained by taking five swings instead of three. 
Example : — 

READINGS. 

LEFT. RIGHT. 



S.8 
9.2 
9.4 



11.7 
11.4 



Average 9.13 11.55 

Zero 10.34 



Exercise 10. — To determine the sensibility of the balance. 

The sensibility for any load is defined as the difference 
in reading on the scale produced by placing an overweight 
of one mg. in one pan, when the balance has a weight in 
each pan equal to the given load. 

Place the weight for which the sensibility is desired, 
say 10 g., in each pan, and find the position of rest of the 
pointer by the method of swings described above. Sup- 
pose this was 10.32. Now, with the rider add 2 mg. to the 
weight in the right-hand pan and again find the point of 
rest. Suppose it to be 7.52. The difference of these two 
readings divided by 2 will give the difference for 1 mg., or 
the sensibility. 

(10.32 - 7.52) /2 = 1.4 divisions. 

In this way find the sensibility for loads of 0, 10, 20 N 
50, and 100 grams. Tabulate the results and plat a curve. 
• general directions, p. 5.) 

E.rrrcise 11. — To perform a -weighing. 

Place the object to be weighed in the left-hand pan 
and place weights in the other, and adjust the rider until 



MEASUREMENT OF MASS. 23 

the point of rest, determined by swings, is not more than 
one or two divisions from the zero. Then determine the 
sensibility for this load by adding or subtracting 2 mg., as 
in the preceding exercise. If a curve of sensibility for 
this particular balance has been drawn, the value for this 
load may be determined by interpolation from the curve. 
The difference between the zero and the above point of 
rest, divided by the sensibility, will give the amount in 
milligrams to be added or subtracted to bring the pointer 
to zero. 

Example : — 

Suppose that with a weight of 23.465 g. the point of 
rest is 7.3 and that the zero is 10.12. If the sensibility 
for this load is 1.3, the weight, which must evidently in 
this case be subtracted, is — 

(10.12 - 7.3) / 1.3 = 2.17 mg., 

and the true weight is — 

23.465 - 0.00217 = 23.46283 g. 

Give result to hundredths of a mg. 

Double weighing and the determination of the ratio of 
the arms of a balance. Weighings are usually performed 
on the supposition that the arms of the balance are exactly 
equal, a condition which is never perfectly fulfilled. The 
error due to the inequality of the arms may be eliminated 
by weighing the body first in one pan, and then in the 
other, and taking the mean of the two values. 

The necessity of making double weighings on any one 
balance may be avoided by determining once for all the 
ratio of its arms. This ratio can be calculated from any 



24 PHYSICAL MEASUREMENT. 

double weighing. Suppose an object when weighed in the 
left-hand pan has an apparent weight, IF, and in the right- 
hand pan an apparent weight,* W\ and that the length of 
the corresponding arms are L and R. From the principle 
of moments, if m is the true weight of the body, — 

1) RW = Lm 

2) L W = Em ; 
multiplying, 

m= VWW, 

which for a small variation of the arms is nearly — 

m=(W+ W) / 2. (See page 123.) 
Dividing Eq. 1 by Eq. 2, we have the ratio of the arms : 



R / L = VJT'/ W= Vl + (W f -W) / W 

= l + ( w r - W) /2 W, nearly. 

Letting 1 + (W 1 - W) j 2 W= c, then c will be a con- 
stant, since R / L is a constant, and the true weight, m, of 
any body will, from Eq. 1, be the apparent weight W times 
the correction constant <?, or 

m = Wc. 

Weight in vacuo. According to the principle of Archi- 
medes a body weighed in a fluid is buoyed up by a force 
equal to the weight of the fluid which the body displaces. 
If a substance more or less dense than brass be weighed 
with brass weights, it will displace a smaller or greater 
volume of air than is displaced by the weights. Hence 
the true weight of a body can only be found by finding 
its weight in vacuo. This can be done as follows: — 

Let the weight of the body in air be w, its density 6?, 
the density of the weights 8, and the density of the air X. 

Volume = Mass / Density. 



DENSITY. 25 

Then, if m is the true weight of the body in vacuo, — 

m (1 - A/ cl ) = w (1 - A / 8) ; or, 

m = w(l- A/S)~ (1 -k/ d). 

Since A is very small compared with c? or 8, we may 
write — m = w (1 + A / d - A / 8). (See page 123.) 

This correction is often of great importance, since the 
air sometimes introduces an error amounting to 0.006 of 
the whole weight. The values of A for different temper- 
atures and pressures may be found in Table 9. Generally 
the mean value may be taken as 0.0012, and with this 
value for A the corrections for brass weights may be taken 
directly from Table 11. 



DENSITY. 



The density of a body is defined as the ratio of the 
mass to the volume, that is, the mass of unit volume. In 
the C. G. S. system the unit of mass is the gram, and the 
unit of volume the cubic centimetre. The gram is defined 
as the weight of a cubic centimetre of water at 4° C. 
Hence density or specific gravity may be defined in these 
units as the ratio of the mass of the substance to the mass 
of an equal volume of water. 

(a). Correction for Temperature. — Since the density of water is 
unity at 4° C, in all accurate determinations involving comparisons with 
the mass of water at other temperatures we must calculate from the 
mass of the water at the given temperature the mass of an equal volume 
of water at 4° C. This is done by dividing the mass at the given tem- 
perature by its density at that temperature as given in Table 6. 



26 PHYSICAL MEASUREMENT. 

(b). Correction for Loss of Weight in Air. — In all accurate deter- 
minations the reading of the barometer should be taken, and the weigh- 
ings reduced to weight in vacuo. (See p. 24.) 

Exerd%e 13. — To determine the density of a regular solid 
from its mass and volume. 

Weigh the body, and find its dimensions as accurately 
as possible. Then if m is the mass, d the density, and v 
the volume, — 

d = m / v. 

Exercise 14 a. — To determine the density of a liquid -with 
the specific gravity bottle. 

The specific gravity bottle is a light glass bottle with 
a ground glass stopper, which is usually provided with a 
small hole to prevent the retention of air-bubbles in the 
bottle when it is filled with a liquid. If the stopper is 
not provided with a hole, the retention of air may be 
avoided by filling the bottle brimful of the liquid and tilt- 
ing the stopper as it is inserted. 

Weigh the bottle empty, being sure that it is perfectly 
dry, fill it with distilled water, and note the temperature of 
the water. Insert the stopper, and invert the bottle to see 
that no air-bubbles are present. Then remove all water 
from the outside with filter-paper, and weigh. Next empty 
the bottle, and rinse it with alcohol, which unites with the 
water, and then rinse with ether, which unites with the 
alcohol. This is volatile, and quickly evaporates and 
leaves the bottle clean and dry. Now fill the bottle with 
the given liquid, and repeat the operations performed with 
ili«' water. Determine the mass of both substances by 
reduction to weight in vacuo, which in this case amounts 
merely to adding the weight of an equal volume of air to 



DENSITY. 27 

the weight of the two liquids. Now calculate the mass of 
an equal volume of water at 4° C, and substitute in the 
general formula, d = m / v. 

!>• — To determine the density of a solid with the specific 
gravity bottle. 

Determine the internal volume of the bottle as in the 
preceding exercise. Weigh the body in air. Then place 
it in the bottle, and fill the remaining space with water. 
Remove any air-bubbles that may adhere to the body by 
shaking it about in the bottle before the stopper is in- 
serted. Weigh the contents of the bottle, and deduct its 
mass from the mass of the substance plus the mass of the 
water when the bottle is full. This will give the mass of 
the water displaced, or the volume of the body. Substi- 
tute as before in the general formula for density. 

Give results to three places of figures. 

Note. — Beyond this point give all calculated results to three places 
of figures unless otherwise directed. 

DETERMINATION OF THE DENSITY OF SOLIDS BY THE 
METHOD OF IMMERSION. 

The principle of Archimedes furnishes a convenient 
method of determining with great accuracy the volume 
of a solid, no matter how irregular its shape. The weight 
of the body is first taken in air, and then when suspended 
by a fine wire in water. The difference in weight gives 
the volume of the body. The weight of the body in 
water is found by deducting from the weight of the body 
and wire in water the weight of the wire alone immersed 
to the same depth as before. The values thus obtained 
for the 'mass and volume of the body are then substituted 
in the general formula for density. 



28 PHYSICAL MEASUREMENT. 

Exercise 15 (fl). — Find the density of copper or lead by 
the method of immersion. 

Caution. Be careful in all eases to see that no air-bubbles 

adhere to the immersed body. These may be removed by means 
of a small wire or pencil brush. 

(7>). — Find the density of cork or paraffine. 

If the solid is lighter than water, it will be necessary 
to add a sinker. The volume of this sinker, as well as 
that of the immersed part of the wire, must be determined 
and deducted from the total volume. 

Solids soluble in water should be weighed in some 
liquid in which they are insoluble. From the known 
density of the liquid the volume of the immersed solid 
may be calculated. Salts in the form of crystals may be 
immersed in a saturated solution of the same salt. 

Exercise 16. — Find the density of alcohol or a saturated 
solution of copper sulphate by the method of im- 
mersion. 

Find the loss of weight of the same body, first in the 
liquid and then in water by immersing a glass ball or 
stopper suspended by a platinum wire. 

The corrections for the wire and for the temperature 
of the water should be made, and the weights reduced to 
weight in vacuo. 

VAPOR DENSITY BY VICTOR MEYER'S METHOD. 

In this method a quantity of the solid or liquid whose 
vapor density is sought is weighed in a small bottle, B 
(Fig. 9), which is provided with a ground glass stopper. 
The glass tube, r l\ is inside a bath, H, which contains a 
Liquid of considerably higher boiling-point than the sub- 



DENSITY. 



29 



stance under investigation. A graduated tube, Gr, filled 
with water is connected with T by the small tube, T . 
The determination is made as follows : The bath, H, 

is heated until a constant tem- 
perature is assumed; i.e., until 
no more air-bubbles come up 
through the water in V from 
the tube r, which at first is not 
covered by the graduated tube, 
Gr. The cork, 6 7 , is removed, 
and the bottle containing the 
substance is dropped in and 
the cork at once replaced. At 
the same time the graduate is 
placed over r. The bottle falls 
on a bed of asbestos, A, the substance at 
once assumes the gaseous state, blowing 
out the stopper of the bottle, and expel- 
ling from T a volume of air equal to its 
own volume. This air enters the gradu- 
ate, Gr, expelling some of the water. The 
number of c.c. of displaced water is read. 
Let — 

this volume be v, 

the weight of the substance, m, 

the temperature of the water, t, 

the pressure of the displaced air in cm. of mercury, P, 
the height of the water column, h, 

Besides the pressure of the air in (7, there is the pres- 
sure due to the water vapor. P is the sum of these two. 
The density — 




Fig. 9. 



d = 



mass 
volume 



30 * PHYSICAL MEASUREMENT. 

Since all ^ascs have the same coefficient of expansion, 
a = 0.00367, if the gas lias a volume v at a temperature 
t and a pressure I\ for a temperature 0° and a pressure of 
76 cm. of mercury, the volume — 



1 + 0.00367 • t 76 



Considering the presence of the water vapor, it may be 
shown that, instead of the true coefficient of expansion 
0.00367, 0.004 should be used. If the density is to be 
given in respect to air as unity, since 1 c.c. of air at 
normal temperature and pressure weighs 0.00129 g., this 
density- ^ _ m 76 1 + 0.004 • t 

v'- 0.00129 ~~ v P ' 0.00129 

The pressure, P, is the barometric pressure, 5, less the 
pressure of the column of water, A, expressed in cms. of 
mercury. 

P = B h 



13.6 
Exercise 1 7 . — To determine the vapor density of ether. 

Dry the tube, T 7 , thoroughly by washing with alcohol 
and ether. Place a little dry asbestos in the bottom, and 
immerse in the water bath, H. Heat this to boiling, and 
wait till the temperature in T becomes steady. Then drop 
the little bottle containing the weighed quantity of ether 
into T, and as nearly as possible at the same time place 
the graduate, (?, over r. Calculate the density as above. 



GRAVITY. 31 



GRAVITY. 



Exercise 18. — Determination of the acceleration of gravity 
■with the simple pendulum. 

The time of oscillation, £, of a simple pendulum is ex- 
pressed by the formula — 

V 9 
where I is the length, and g is the acceleration due to 
gravity. Then — 

q = tt" — 
t 2 

and I and t are to be determined. 

The method of coincidences is a convenient method for 
determining t. An electric circuit containing a sounder 
is arranged so that it is completed only when the pendu- 
lum of the clock and also the pendulum to be compared 
with it are both in the vertical position. The plan of the 
circuit is shown in Fig. 10. A convenient form of mer- 
cury contact is illustrated in Fig. 10, a. The narrow 
slots, s, are kept full of mercury from the larger reser- 
voirs, r. The oxide formed at the contact is knocked 
aside by the wires, and falls into the large grooves, g^ 
thus keeping the surface always clean. This double 
contact makes possible the use of electric connections 
with a thread instead of a wire suspension, thus making 
the pendulum approximate very closely to a simple one. 
To prevent a twisting motion a bifilar suspension is used. 
To start the pendulum with a steady motion, it may be 
drawn aside by grasping the projecting wire, TF(Fig. 10 5). 



VII TB ICu I L M Eu 1 8 U REM EN T. 



Set both pendulums in motion. The sounder will be 
heard when both pass through the mercury contacts at the 
same instant. This takes place as often as the longer pen- 
dulum loses one oscillation. Take down the minute and 
the second of each click for five or ten minutes, and find 




• 






• 

r 


( 


) 


( 


3 ) 



CL 



Fig. 10. 

the average interval between them. If the sounder at 
any time should fail to work, a double interval may be 
obtained, and should be so reckoned. When the swings 
become very short a scries of clicks may be heard at each 
contact. Their average is the time of coincidence. 



GBAVITT. 33 

By counting the number of times the simple pendu- 
lum swings in one minute, it may be found whether it 
is shorter or longer than the clock pendulum, which is 
supposed to beat seconds. If the interval between two 
successive coincidences was found to be 2-i seconds, and 
the simple pendulum vibrated slower than the clock pen- 
dulum, the former must have made 23 beats while the 
latter made 24, or its time of oscillation is \% sec. 

The pendulum is placed in front of a scale graduated 
upon a mirror. The length of the pendulum is the aver- 
age of the distances from the point of suspension to the 
top and bottom of the ball. If the observation be made 
with the eye in such a position that the edge of the ball 
exactly coincides with its image in the mirror, there will 
be no error of parallax. 

The above formula for t is only true for a very small 
arc. The error for an arc of 5° is .0005, and may there- 
fore be neglected. This corresponds on a pendulum a 
little over a metre long to about 12 cm. This amplitude 
should not be exceeded in the experiment. 

Give results to four places of figures. 

For a pendulum about a metre long the ball should 
not exceed a diameter of 2 cm. Otherwise a correction 
must be made for the moment of inertia of the ball. For 
a ball of radius, r, the corrected length of a pendulum 
whose apparent length is ?, becomes 

o I 

See Stewart & Gee, vol. i., p. 227. 



84 PHYSICAL MEASUREMENT. 



ELASTICITY. 



Exercise 10. — To determine the modulus of elasticity of a 
steel wire by stretching. 

Young's modulus of elasticity is defined as the force 
which would be required to double the length of a 
wire of unit cross-section. The units commonly taken 
are kilograms and square millimetres. The modulus, U, 
is then — ^ F 

la 

Where L equals the length of the wire in mms., 
a equals the cross-section in mins,, 
I equals the elongation in mms., 
F equals the force in kgs. 

The quantities to be measured are the length and 
cross-section of the wire and the elongation. The elon- 
gation may be measured conveniently by means of the 
filar micrometer, or by the use of the optical lever, which 
is an application of the mirror and scale. This principle 
is of such importance and of such frequent application 
that a general description of it will be in place here. 

The Optical Lever. — According to the law of reflec- 
tion, the angle made by the reflected ray with the perpen- 
dicular to the reflecting surface is equal to the angle made 
by the incident ray with the perpendicular. An observer 
looking through the telescope at the mirror (see Fig. 11), 
in the position aCb, will see the scale division reflected. 
The perpendicular, 7 J , the incident ray, I, and the reflected 
ray, R, arc all in the same plane. If the mirror be turned 



ELASTICITY. 



through an angle, a, to the position a!Cb', the incident ray 
will come from the scale division, 10, and make an angle, 
2 a, with the reflected ray. 



Here the tangent of 2 a 



E 



where n is the number of scale divisions ior a deflec- 
tion, a, of the mirror, and R the distance from the mirror 
to the scale expressed in the same units as n. If R is 






f 

-4- 



-Ac 



Fig. 7 7. 



large compared with ??, the tangent and arc may be taken 
as equal, and we may write — 

, n 

tan a = . 

2R 

A convenient form of optical lever for our present pur- 
pose consists of a brass bar, B (Fig. 11 a), heavy enough 
to keep its position upon three needle-points, two of which, 
attached to the support, S, form the fulcrum, F, of our 
\ever, the third being attached to the heavy brass ring R' . 
The bar is kept in a constant position on the points by 
means of shallow conical holes at H, P, P' . To prevent 
lateral motion, the wire passes loosely through a small hole 



36 



PB rSK 'AL MEASUREMENT. 



in the plate, M. To the lower side of if are fastened two 
pairs of small rods, between which plays the cross-bar, 5, 
thus preventing rotation. If the end of the lever move a 
distance JIJI' = /, and the ray of light a distance w, calling 




Fig. 11a. 



r the length of the short arm of the lever, R the distance 
from the mirror to the scale ; then, by geometry, — 



B::l:r. 



To Adjust the Telescope and Scale. — First approach 
the mirror, and, looking into it, determine in what direc- 
tion it is pointed, and adjust it so that it shall face as 
nearly as possible in the desired direction. Then, stand- 
ing by the telescope, move the eye, or the scale, or both, 
until the reflection of the latter is caught sight of. Then 
place the telescope in the position from which the reflec- 
tion was seen, and, sighting over its top, point it at the 



ELASTICITY 



37 



mirror. When the mirror is found, cnange the ±ocus until 
the scale (which is optically twice as far away as the 
mirror, and hence demands a shorter focus) is distinctly 
seen. Next, to prevent parallax, adjust the cross-wires by 
sliding the ocular, o (Fig. 11), in or out until they are 
plainly seen, and do not seem to move in reference to the 
scale divisions when the eye is slightly moved. 

Measure the length of the wire, Z, from the top of the 




Fig. 12. 



ring to the point of support. Measure the diameter of 
the wire, d, from which compute the area of the cross-sec- 
tion, a. Measure r and i?, and, by the method described 
above, find the scale readings for an addition of 1, 2, 
and 3 kgs., taking the readings both as the weights are 
added and again as they are removed. See that the read- 
ing after the weights are removed is the same as at first, 
so as to be sure that the limit of elasticity has not been 
passed. R should be measured correctly to 10 mm., L 
to 1 mm., r to .1 mm., and d to .01 mm. The value of a 



38 PHYSICAL MEASUREMENT. 

scale division should be verified by measuring the length 
o( the paper scale. The scale readings should be taken in 
as rapid succession as is consistent with accuracy, to avoid 
changes in the length of the wire due to temperature vari- 
ations. 

MODULUS OF ELASTICITY BY BENDING. 

Exercise 20. — Determine the modulus of elasticity of a 
steel bar. 

If a rectangular bar (see Fig. 12) of width 6, and ver- 
tical thickness A, be supported upon two knife edges at a 
distance, Z, from each other, and a weight, w, be suspended 
from its middle point, this point will be lowered by an 
amount I. Then, if JE be the modulus of elasticity, it may 

be shown that — ,. _ 

j2 = 1 ivL 3 1 

4TJbh 3 ' 

Measure b and h with the micrometer gauge at inter- 
vals of 10 cm. throughout the length of the bar. Measure 
L with a steel tape or two metre sticks. Take the values 
of I for iv = 1 kg., 2 kg., and 3 kg., taking readings both 
as the weights are put on and taken off. A convenient 
means of measuring I is a micrometer screw attached to a 
fixed support extending over the middle of the bar. The 
instant of contact of the screw may be accurately deter- 
mined by observing the image of some object in a small 
mirror, one end of which rests upon the bar and the other 
upon an adjacent support. 

ELASTICITY OF TORSION AND DETERMINATION OF 
MOMENTS OF INERTIA. 

The torsional pendulum, like the simple pendulum, is 
use of harmonic motion. Its period is, therefore, — 

1 For the derivation of this formula, which involves the use of the 
calculus, the student is referred to Kohlrauseh, or Stewart & Gee. 



ELASTICITY. 39 

Resistance 



-V 3 



Force 

In this case the resistance is evidently the moment of 
inertia, I, of the weight, and the force tending to bring the 
pendulum to a position of equilibrium is the directive 
force, T, which is defined as the torsional elastic force of 
the wire, which is, of course, equal to the deforming force 
when the angle of torsion is an unit angle. 

i* 



(A) * = , y 

The modulus of torsional elasticity, /x, often called the 
coefficient of simple rigidity, is proportional to the ratio 
of twisting force to the resulting angular displacement. 
We learn from works on elasticity that the unit angular 
displacement is — 

. A JL L 

* = — r> 

where I is the length, and r the radius of the wire. 

From (A) T = ~; 

therefore, for unit angle, ^ = 1= * ; and, 

flt'T 

( Bi ) /* = —?hr > 

where the units used are the kilograms of mass and milli- 
metres. If the kilogram be taken as a unit of weight, not 
mass, we shall have, — 

(B 2 ) n = _£- = 0.0006405 — 
v ; 9810 t 2 r± 

kilograms of weight per sq. mm. 

* This formula is only strictly true when the arc is infinitely small. 
Corrections for arcs of any size are given in Table 30. For the theory, see 
Kohlrausch. 



40 PHYSICAL MEASUREMENT. 

MOMENTS OF INERTIA. 

The moment of inertia, i, may be determined either 
theoretically, from the dimensions of the body, or experi- 
mentally. 

The moment of inertia of a slender bar of mass wi, and 
Length /, about an axis through its centre of gravity at 
right angles to its length is — 

(d) J=— . 

v J 12 

For a cylinder of radius r, and length Z, about an axis 
identical with the axis of the cylinder, — ■ 

(oo /-=£. 

For an axis at right angles to the axis of the cylinder 
passing through the centre of gravity, — 

(C.) *-.«•(£ + £ 

For a sphere of radius r, about an axis through the cen- 
tre, — o 

(C 4 ) I=^mr 2 . 

For a rectangular parallelopipedon of sides Z, 5, and A, 
about an axis parallel to A, — 

l 2 + b 2 



(C 5 ) I = m - 



12 



If 7, be the moment of inertia of a body of mass ???, about any axis 
passing through the centre of gravity of the body, I a the moment of 
inertia about an axis parallel to the first, and at a distance a, from it, 
it may be proved that — 

I a = I -f- via 2 . 

To determine / experimentally, use may be made of 
the principle of the torsion pendulum. (See p. 38.) 



ELASTICITY. 



41 



t = 7T 1 / — 



/T 



For any other moment of inertia I', obtained by adding 



weights, — 



ir 



f = 7rJ—; 



HI' = I + I x , where i" x is the moment of inertia of the 
added weights, then 

(D) I = I x f2 . 

TIME OF A PERIODIC MOTION. 

The time of oscillation, £, may be determined by any of 
the following methods : — 

1. By observing the number of seconds required for 
the pendulum to make 10 oscillations, and dividing this 
number by 10, the result will be given to 0.1 sec. By 
observing the time of 100 oscillations t may be found to 
0.01 sec, etc. The observation can be most accurately 
made when the pendulum is at the middle point of its 
swing, when it is moving at its maximum velocity. 

2. By the method of coincidences explained on p. 31. 

3. By the method of middle elongations. 




Fig. 13. 



Let Figure 13 represent cliagrammatically the succes- 
sive positions of a pendulum. The successive passages 



42 PHYSICAL M K. LSU E EM KNT. 

are represented by the figures 1, 2, 3, etc. Midway be- 
tween any two passages the pendulum comes to rest at 
an extreme position, i\ called its elongation. The average 
of any two successive passages will give the time of elon- 
gation occurring between them. The average of the time 
of a series of 10 passages will give the time of the fifth, or 
middle, elongation of the series. If a second series of 
observations be taken after time enough has elapsed for 
the pendulum to have made about 50 oscillations, the 
interval between the middle elongations of the two series, 
T, divided by the number of oscillations, n, that took place 
during the interval, will give with great accuracy the time 
of one oscillation. The number of oscillations may be 
conveniently determined as follows : Find an approximate 
value, t\ for t, by dividing the interval between the first 
and ninth passages in any series by eight. 

Then, £ = n\ 

if 

Since there must have been a whole number of swings 
during the interval T, any divergence from a whole num- 
ber is due to the error in t\ and n may be taken as the 
nearest whole number to n\ whence — 

t = L 

n 

% An even number of oscillations is taken, so that no 
error may be introduced from the point of passage not 
being exactly half-way between two elongations. 

Methods (1 ) and (2) can be advantageously used for 
rapid vibrations, method (3) for slower ones. 

If a telescope with eross-hair is used for observing the 
passages, the time of passage may be estimated to tenths 



ELASTICITY. 



13 



of a second by the method described in the following 
exercise : — 



Exercise 21. — (a) To find the moment of inertia, I, of a 
body and (J) the modulus of torsion, n, of the sus- 
pending wire. 

The wire should be clamped firmly at its upper end to a 
fixed support. Upon the weight which is rigidly fastened 
to the lower end of the wire is a vertical mark, the passage 
of which can best be observed by means of a reading tele- 
scope provided with cross-hairs. Find the time of oscilla- 
tion £, by method (3). The time of each passage of the 
mark past the cross-hair should be noted and recorded in 
hours, minutes, and seconds and tenths of a second. First 
practise counting seconds with the clock till it is possible 
to keep counting at the same rate for several seconds 
without looking at the clock. 
As the mark approaches the 
point of transit, look through 
the telescope while continu- 
ing to count. Suppose at 2 
hr. 15 min. 21 sec. the mark 
is approaching the cross-hair 
and at 25 sec. has already 
passed it, then the fraction 
of the 21th second at which 
the transit occurred may be 
estimated from the relative position of the mark with ref- 
erence to the cross-hairs at the 21th and 25th seconds. 
The transit illustrated in Fig. 11 occurred at 2 hr. 15 min. 
21.6 sec. 

Having determined £, acid a known moment of inertia, 
I v and determine the new time of oscillation, t'. If the 




Fig. 14. 



44 PHYSICAL MEASUREMENT, 

added weight consists of two equal cylinders of radius p, 
placed on opposite sides of the axis of rotation and with 
their centres at a distance 11 from it, then, — 

J, = m(£' + £,*)*, 

where m is the mass of the two weights. The moment of 
inertia, V, of the original weight is from (D) — 

t 2 



Ix 



t ,2 -t 2 ' 



(7>) To find ?i, the modulus of torsional elasticity, 
measure the length /, and radius r, of the wire, and substi- 
tute these values and the value for / obtained above, in 



n= 0.0006405- — . 
t 2 r* 



According to the theory of elasticity, n is about § Young's 
Modulus, E. 



CAPILLARITY. 



Exercise 22. — To determine the capillary constant of -water. 

The capillary constant, or coefficient of surface tension, 
is defined as the weight of liquid raised above its natural 
level per unit of length of the line bounding the surface 
of the liquid raised. In the case of a tube the weight is 
hvr&dff, and the line bounding the surface is 2?rr, where h 
is the height of the capillary column, r its radius, d the 

* The term j /no- is the value of the moment of the two cylinders 
about their own axes. If the weights were attached by flexible threads, 
this term would dropout, leaving J 1 = ,,ilt' 2 . (See j). 40.) 



CAPILLARITY, 



45 



density of the liquid, and g the force of gravity. The 
capillary constant is, then, — 

a = — — = J h rag. 

To measure r, the radius of the capillary tube. See 
that the tube is perfectly dry. then draw into it a thread 
of mercury about 3 or 4 cm. in length. Measure the 
length of the mercury thread. L and weigh the mercury 
carefully in a watch glass. 



and 



The weight is — 
= b-lr 2 . 



\ 13.55 -V 
where the density of mercury. 8. for 20° C = 13.55. 

To measure h. first clean the tube with a little nitric 
acid and afterwards 
with water. The 
height may be meas- 
ured by means of 
the apparatus shown 
in Fig. 15. The 
capillary tube is in- 
serted in a hole at 
the centre of the 
little board which 
rests upon the top 
of a crystallizing 
d i s h. From the 
lower side of the 
board project two 
points, j9, j>\ to a 
distance of 2 cm. from the top of the board. A scale is 
attached to the top of the board from which the height 




Fig. 15. 



40 PHYSICAL MEASUREMENT. 

of the column is read. The scale is best ruled on a mir- 
ror. The liquid is poured into the dish until its surface 
just touches the points, p, p'. The final adjustment can 
be made by means of the pipette, P. 

Before taking the reading, lower the tube and raise it 
again, to be sure that it is wet as high as the column will 
rise. 

Give results to two places of figures. 

The foregoing method is adapted to determining the 
capillary constant of substances which adhere to glass, as 
is shown by their wetting the glass. For substances like 
mercury, which do not wet the glass, the coefficient of sur- 
face tension may be determined from the height of drops 
of the substance resting upon a level surface to which they 
do not adhere. 

Exercise 23. — To measure the surface tension of mercury. 

Measure A, from the level of the widest part of a 
drop * of mercury to its top, with the filar micrometer, — 

a = \ dh 2 ^ 
which is for mercury — 

a = 6.78 A 2 . 



VISCOSITY. 



JExercise 24. — To find the coefficient of molecular friction or 
viscosity by the flow through capillary tubes. 

The coefficient of viscosity, 77, is defined as the force 
which tends to prevent the motion of a sheet of unit area 

* The drops should be about 1 cm. wide. 

ee Wiillncr. Experimental Physik, vol. i., p. 332. 



VISCOSITY. 



47 



of the fluid moving with unit 
velocity at a unit distance from 
the side of the containing tube, 
where, of course, the velocity 
is zero. It is evident that, if 
a liquid flows through a tube, 
the volume passing through is 
proportional to t, the time of 
flow, p, the pressure forcing it 
through, and inversely propor- 
tional to /, the length of the 
tube, and 77, the coefficient of 
viscosity. By the calculus it 
may be proved that the flow is 
also proportional to 7rr 4 /8, where 
r is the radius of the tube. 
Then the volume — 

8 v l' 

Here the pressure, p, in abso- 
lute units, equals the height of 
the column of liquid, A, times 
its density, c?, times the acceler- 
ation of gravity, g. Then — 

iti a hd 981 , 

v = T " -fT ' *' 

To find the value of rj 
for water, first clean the 
apparatus (see Fig. 16) 
well by pouring through 
water, alcohol, and then 
distilled water. Fill the 





Fig. 16. 



{^ PHYSICAL MEASUREMENT. 

bulb with the latter, and start it (lowing through the 
tube /. When all the bubbles are out of the tube, hold 
a clean glass dish so as to catch the water for a length 
of time t (about 15 or 20 seconds). Then weigh the 
water and find its volume. Measure the length of the 
capillary tube, h and find /*, which must be taken as 
the average height of the water column at the beginning 
and end of the time, t. The radius, r, is found by meas- 
uring under the microscope the bore of the small piece of 
the tube /, previously broken off for the purpose. 

From this data rj may be found by the above formula. 
The temperature, T, of the water must be taken, as rj varies 
for different temperatures. 

From the above equation, if we have two liquids hav- 
ing the coefficients r j 1 and 772? aR d densities d x , cZ 2 , — 

Therefore, if we have found the value of rj for water, we 
can find the value of the coefficient rj of any other liquid 
of known density by comparing the volumes of the two 
liquids which flow through the apparatus in equal times. 

In this way determine the viscosity of alcohol. 

Give results to two places of figures. 



HEAT. 



CALIBRATION OF THERMOMETERS. 
rdse 25. — To calibrate a thermometer. 

The melting-point and the "boiling-point of water are 
taken as the fundamental points in graduating a ther- 
mometer. 



HEAT. 49 

(a) To determine the zero point. Place the ther- 
mometer in a vessel containing pounded ice which has 
been standing in a warm room long enough to have 
attained the melting temperature. The vessel should 
have a perforated bottom for the escape of the water 
from the melting ice. If the ice was very cold, take a 
second reading some time after the first, to make sure 
that it had reached the melting temperature throughout. 

(6) To determine the 100° point. That temperature 
at which the vapor pressure of the liquid becomes equal to 
the normal pressure of the atmosphere (760 mm.), and at 
which boiling sets in, is defined as the boiling-point. The 
liquid is placed in a flask provided with a cork pierced 
with two holes, one for the escape of the vapor, and the 
other for the thermometer. The bulb of the thermometer 
must be a cm. or more above the surface of the liquid, as 
the temperature sought is that of the vapor, and under 
certain conditions the liquid itself may become somewhat 
hotter than the true boiling temperature. Heat the liquid 
over a Bunsen burner until the thermometer ceases to rise, 
wait ^ minute, and note the temperature, t. The water 
must not be boiled too vigorously, else the vapor pressure 
in the flask will exceed the pressure of the outside air, 
and the apparent boiling-point will be too high. There 
are two corrections to this reading. 

I. The boiling-point is raised 0.0375° C. for each mm. 
increase in the height of the barometer. Then, to find the 
boiling-point at the standard pressure (760 mm.), there 
must be added to £, 0.0375 (760 — 6), where b is the height 
of the barometer in mm. at the time of the experiment. 

II. The reading t is too low, since a part of the mer- 
cury thread is out of the vapor. To find the temperature 
of the part of the thermometer above the cork, place the 



50 PHYSICAL MEASUREMENT. 

Lulb of a second thermometer against the first, and note its 
temperature, /'. Then, as the relative expansion coeffi- 
cient oi mercury in glass (that is, the difference between the 
coefficients of expansion of mercury and glass) is 0.000156, 
it' the whole mercury column had been in vapor, the read- 
ing* would have been higher by 0.000156 ■ a ■ (£ — £'), where 
a is the length of the mercury column above the cork ex- 
pressed in degrees. The true boiling-point is, then, — 

T = t + 0.0375 (760 - b) + 0.000156 - a • (t - f). 

(e) The most convenient, though not the most accu- 
rate, way to calibrate the intermediate portions of the ther- 
mometer, is to compare it with a standard thermometer. 
The thermometers to be compared are fastened together 
with rubber bands, so that they dip to the same depth in a 
beaker of water or oil, oil being required if the calibration 
is to go above 100°. It is now only necessary to compare 
the thermometer with the corrected readings of the stan- 
dard thermometer. Readings should be taken every ten 
degrees from 0° to 100°, and results given to T l - 6 °. 

For absolute calibration of a thermometer, see Kohl- 
rausch. 

In all work in heat, when more than one thermometer 
is used, and accurate readings are required, the thermome- 
ters should be compared as above, and all their readings 
may then be given in terms of one. 

Exercise 26. — To determine the boiling-point of a liquid. 

Proceed as in determining the boiling-point of water, 
Exercise 25 (5). 

Exercise 27. — To determine the melting-point of a solid. 

Draw out a portion of a small glass tube to a diameter, 
inside, of about 1 mm., and bend the tube as shown in 



HEAT. 



51 



/? = 



*(*'-*)' 



Fig. 17. Melt a portion of the substance to be examined, 
and draw it into the tube. Fasten the tube 
to a thermometer by means of two small rub- 
ber bands, and immerse the lower part in 
water. Heat the water slowly till that por- 
tion of the substance in the capillary part 
of the tube begins to melt, note the temper- 
ature, remove the heat, and again note the 
temperature at which the substance solidifies. 
The moment that the change of state occurs 
may be known generally by the change of 
color or transparency which takes place in 
the substance. 



COEFFICIENT OF EXPANSION. 

The coefficient of linear expansion is de- 
fined as the change in length per unit of 
length for a change of temperature of one 
degree. The coefficient of cubical expan- 
sion is the change of volume per unit of 
volume for a change in temperature of one 
degree. The coefficient of cubical expansion 
is approximately equal to three times the 
coefficient of linear expansion. 

Exercise 28. — To find the coefficient of linear 
expansion of a metal bar. 

If I is the length of the bar at a temper- 
ature £, and V its length at a temperature t\ 
then the coefficient of linear expansion is — 

V - I 



Fig. 17. 



The bar is suspended in a tank of water, one end resting 



52 



/'// YSICAL MEASUREMENT. 



against a fixed arm, .1 (Fig. 18), the other against a mova- 
ble arm, A\ which is held firmly against the bar by means 
o( the counterpoise, C. To the axis of the movable arm 
is attached a mirror, w, by means of which the change in 
length of the bar can be measured by the method de- 
scribed in Exercise 19, p. 34. The temperature of the 
bar should be observed by means of three thermometers 
placed with the bulbs near the bar at equal distances 
along its length. The water should be thoroughly stirred 
so as to keep the readings of all three thermometers the 
same. Heat is applied by means of a row of burners 





\ 


?) 






/ 


"1 

r 


\ 




A' 






A 






s - 1 


-— _— - — - — : 


_r:~ -_— r — — — 


— L-f^-L-!— — 


^"n 




r 


nc 


~~x 


~? 




n 






F/gr. 75. 



placed below the tank. It is Avell not to heat the water 
too rapidly. Take readings at intervals of 10° from the 
initial temperature to 60° or 70°. 

Another form of apparatus which is very easy to make 
is shown in Fig. 19. A rod A, which is clamped to a 
heavy retort stand, is passed through a metal beaker and 
soldered firmly to it. A point at the lower end of the rod 
B, whose expansion is to be measured, rests in an indenta- 
tion in A. A similar point at the top of R fits into the 
leg of the optical lever, M. The remaining two legs of the 



HEAT. 



53 



a 



In an equation of this form 
the constants V and /3 may be 
accurately determined from 
a series of corresponding ob- 
servations of I and t — t' . 



B 



lever rest in the indentations in the support B. 
J, of asbestos prevents radiation. 

This problem affords a 
good example for the applica- 
tion of the method of least 
squares to the determination 
of a constant from a series of 
observations. If the length 
of a bar at any temperature 
t' be l\ then the length at 
any other temperature t is — 



A jacket 



Q 



Fig. 19. 



A 



JExercise 29. — To determine the cubical expansion of glass. 

rFill a clean glass bulb (Fig. 20), with an open- 
ing about 1 mm. in diameter, with mercury by means 
of a fine pointed pipette, removing the meniscus by 
means of a card. Find the weight of mercury re- 
quired to fill the bulb at the temperature of the 
room. Call this weight w and temperature t. Im- 
merse the bulb, except the extreme point, in water, 
and heat to a temperature, t 1 ', about 20° higher 
than t. Some of the mercury will be expelled from 
11 the bulb. Cut off the projecting meniscus as be- 
fore, dry the bulb with filter paper, and determine 

Fig. 20, ' J l L 

the weight of mercury, u\ The loss of weight di- 
vided by the original weight, w, times the difference of 



54 PHYSICAL MEASUREMENT. 

temperature will give the difference of the expansion of 
mercury and glass for one degree. Then, if 0.000182 be 
the cubical expansion of mercury, the cubical expansion 

of glass is — , 

a = 3 p = 0.000182 



w(t' - t)' 

where /? is the linear expansion of glass. 

Find 3 values of a for intervals of 20°. The weigh- 
ings must be made with the greatest possible accuracy to 
tenths of a milligram. 

SPECIFIC HEAT. 

Specific heat is defined as the number of units of heat 
required to raise one gram of the substance in question one 
degree. 

Exercise 30. — To determine the specific heat of a solid by 
the method of mixtures. 

First find the weight, M, of the body, in a finely divided 
condition, and also the weight, m,, of the calorimeter, this 
being visually a thin metal beaker. Then weigh out 
enough water in the calorimeter to a little more than cover 
the bulb of the thermometer, and call this weight m. The 
water should be taken at a temperature about as much 
below the temperature of the room as it is expected that 
it will be above it at the close of the experiment. This is 
to eliminate the effects of the radiation of the calorimeter. 
The temperature, £, of the water should be noted to 0.1°. 

Heat the substance to a temperature T, in a test-tube 
placed in the mouth of a boiling-flask. The mouth of 
the tube is closed by a cork, through which runs a ther- 
mometer,* the bulb of which should be surrounded by 

* The two thermometers should be compared before work is begun. 



HEAT. 55 

the substance. "When the thermometer has ceased to rise, 
remove the test-tube, and drop the body quickly into the 
calorimeter, taking care that no water from the outside 
of the test-tube drops in at the same time. The water 
should then be stirred slightly, the thermometer bulb being 
kept covered all the time, and the highest temperature, r, 
attained noted. The amount of heat which the water has 
received is m (r — £), since the specific heat of water is 1. 
The amount given out by the body is SM (T — r), where 
$ is the specific heat. These two quantities must be 
equal. Then S = m (j - f) j M(T - r). 

But the calorimeter and thermometer have also been 
heated, and to m must be added the water equivalents of 
these. By this is meant the amount of water which would 
require the same amount of heat to raise its temperature 
one degree. This of course equals the weight times the 
specific heat of each. It is, for the calorimeter, ??? 1 cr, 
where o- is the specific heat of the calorimeter. 

The water equivalent of the thermometer may be found 
as follows : Heat the thermometer over a flame to a tem- 
perature t 1 (about 50°). Then plunge it into a weighed 
quantity of water m 2 (about 20 g.) at a temperature t 2 . 
Then, if the final temperature of the water is £ 3 , the water 
equivalent e = m 2 (t d — f 2 ) / (t x — f 3 ). 

S = (m + vua- + e) (t — i) / JI (T - r). 

In this experiment the weights should be taken to 
cgs., and the result given to three places of figures. 

Exercise 31. — To determine the specific heat of a liquid by 

the method of mixtures. 

Proceed exactly as hi Exercise 30, using the given 

liquid in place of water, and assuming S, the specific heat 

of the metal used, to be known, find the specific heat of the 



56 PHYSICAL MEASUREMENT. 

liquid, *.* The amount of heat taken up by the liquid is 
%m (t — <) and 

8 = SM(T-t)-(v h cr+e)(r-t) 
m (r — t) 

For numerous other methods of finding specific heat, 
see Kohlrausch. 

LATENT HEAT. 
Exercise 32. — To determine the heat of fusion, f, of a solid. 
Call the specific heat of the substance in the solid 
state s, in the liquid state s'. Heat M grams of another 
substance whose specific heat is S to a temperature T, and 
add it in the calorimeter to m grams of the given solid at 
temperature f, stir till the solid is melted, and note the 
temperature, r, of the mixture before it begins to cool 
slowly by radiation. If the melting-point of the solid be 
t\ the total heat transferred in the operation is, where 
»&!, o> are the mass and specific heat of the calorimeter, 
and e is the water equivalent of the thermometer, — 

MS (T - r) = ms (t' — t) + ms f (r — t 9 ) + (m l( r + e) (t — t) 
+ ™f, 

and the heat required to melt one gram of the substance is 

f= MS(T - r) - ms (t'—t)+ ms'(r - t') - (m l( r + e) (r - t) 

m 

Exercise 33. — To determine the heat of vaporization, V, of a 
liquid. 

Through the ; cork of a florence flask pass a bent glass 
tube (see Fig. -1 ), and connect it to the trap T. Fill the 
flask half full of the liquid, and weigh out m grams of 
the same liquid in a calorimeter of known weight, m v and 
specific heat, or. Ileal the liquid to boiling, and after vapor 

* By the formula fif = (*m+ m** + e) (r — t) / M{ T—r), 



HEAT. 



57 



has been escaping from the tube, £, for some minutes, 
dip the tube under the sur- 
face of the liquid in the calo- 
rimeter, and let it remain 
until the temperature has 
risen about 10°. Then re- 
move the tube. 

The increase in weight 
M of the liquid in the calo- 
rimeter is the mass of vapor 
condensed. If the tempera- 
ture of the vapor was T, the 
original temperature of the 
liquid £, its final temperature 
r, and specific heat s, and the 
mass and the specific heat of 
the calorimeter were m 1 and 
o-, and the water equivalent of the thermometer e, then — 




Fig. 21. 



and 



Mv + Ms (T — t) = (ms -f m l( r + e) (r — t), 

(ms + m x <r + e) (r - t) - JIs (T - r) 
M 



MECHANICAL EQUIVALENT OP HEAT. 

The mechanical equivalent of heat is the work done in 
raising the temperature of one gram of water one degree ; 
in other words, it is the equivalent in work of one calorie 
of heat. 

The method employed is to convert a measured quan- 
tity of work into heat by means of friction, and measure 
the heat. 

The apparatus used is shown in Fig. 22. It consists 
of a whirling table, to the vertical axis of which is rigidly 



58 



PH YSICA L 31 E A S U B EM EN T. 



attached by a non-conducting support a steel cup e x , which 
rotates with the axis. Inside this cup fits a second cup c 2 , 




which contains a weighed quantity of mercury into which 
a thermometer is inserted. The outer cylinder is encased 
in felt to prevent loss of heat by radiation. The inner 
cup is kept from rotating by the arm Z, to the end of 
which a force is applied by weights in a pan which 
hangs from the end of the cord <?, passing over a pulley 
p. The number of revolutions made by the axis is 
recorded by the indicator i. Weights are placed in the 
pan, and the table is whirled at such a speed that the 
friction of the cones keeps the weights in equilibrium. 
The moment of friction is the force of friction F times the 
average radius of the cone r. The moment of the weights 
is ivl. To w we must add w\ the friction of the pulley 
p. Then the total moment of force is — 

I (w _|_ w ') = r F, 
and the force of friction, — 

F=-(w + w'). 
r 



HEAT. 59 

The work done in n revolutions is 

2 irnr (w -f- w ') - = 2 wn (w -\- vf) I. 
r 

Exercise 34. 

Weigh the outer cup, calling its weight m 19 the inner 
cup m 2 , and the inner cup filled with mercury, calling the 
weight of the mercury M. Measure the length of the arm, 
?, and put the apparatus together. By trial find what 
weights will hold the pointer in the centre of the scale 
while the table is turned at a convenient speed. Take 
the temperature t x , of the mercury, whirl the machine 
till the temperature rises 5°, and note the temperature t 2 . 

Determine the water equivalents * of the two cylinders, 
e x , of the mercury, e 2 , and of the thermometer, e z . The 
total amount of heat produced is then 

H= (t 2 - t x ) ( ei + e 2 + e 3 ). 

The friction, w\ of the pulley can be determined as 
follows : The pressure on the pulley is w cos 45° = id / V2, 
and as this acts in both directions, it is 2 w / V2 = to V2. 
If we now put the cord over the pulley, and hang to each 
end a weight \ iv/V2, these will exert a pressure on the 
pulley equal to that exerted by w during the experiment. 
Now determine the overweight, w\ necessary to set the 
pulley in uniform motion, which equals the friction of 
the pulley. Then, if J be the mechanical equivalent 

of heat, — 

JH=2nir (w + uT) I, 
and — 

T 2 n-w (w + vf) I 

* See Specific Heat. 



GO PHYSICAL MEASUREMENT. 



HYGROMETRY. 



Hygrometry has to do with the humidity of the at- 
mosphere. By the absolute humidity of the atmosphere is 
meant the number of grams of water vapor in one cubic 
metre. By relative humidity is meant the ratio of the 
amount of vapor in the air to that required to saturate 
it at the given temperature. The dew point is the tem- 
perature at which the air becomes saturated, and moisture 
is deposited in the form of dew. 

August's psychrometer. 

August's Psychrometer is an instrument designed for 
determining the humidity of the atmosphere. It consists 
of two similar thermometers, the bulb of one of which is 
covered with a muslin envelop which is connected by 
means of a wick with a reservoir of water. The evapo- 
ration of the water from the bulb cools the thermometer, B, 
making it read lower than A. The amount of this differ- 
ence of temperature 8, is proportional to the rapidity with 
which evaporation proceeds, which is in turn inversely 
proportional to the relative humidity of the atmosphere. 

Exercise 35. — To find (ft) the vapor pressure, (6) the rela- 
tive humidity, and (e) the dew point of the atmosphere 
with August's Psychrometer. 

(a) Read the two thermometers. Let the reading of 
the dry one be £, of the wet one £', and let t — t' = S. 
Then, if p 1 he the pressure which would exist in air satu- 
rated with vapor at the temperature t' 9 and barometric 



HYGBOMETBY. 61 

pressure 5, then, by an empirical formula, if t 1 be above 
0°, the vapor pressure at temperature t is — 

p = p ' _ 0.0008 $b. 

The value of p' may be taken from Table 12. For 
indoor observations, where the air is practically at rest, 
as in a small closed room, the factor 0.0008 becomes 0.0012 
(Kohlrausch). If, however, the air be kept in motion 
with a fan, the original factor 0.0008 may be used. 

(6) The absolute humidity h is approximately — 

h = h' - 0.64 8, 

where ti, the humidity of saturated air, is taken from 
Table 12 for the temperature t'. 
The relative humidity is then — 

n = ±. 
K 

where h x is the humidity of saturated air for temperature t. 
(tf) The dew point for any absolute humidity, A, may 
be taken from Table 12. 

DanielPs Hygrometer is an instrument designed for 
determining the dew point. It consists of two exhausted 
bulbs connected by a tube and containing ether. One of 
the bulbs, JL, has within it a thermometer, and has a por- 
'tion of its outer surface covered with a ring of polished 
metal. The remaining bulb, B, is covered with muslin. 
When ether is poured upon this muslin cover, it evaporates 
rapidly, and cools the ether vapor within the bulb, thus 
reducing its pressure, and causing the ether in A to distil 
rapidly into B. The temperature in A is thus reduced, so 
that moisture from the air is deposited upon the bulb. 
The temperature at which this occurs is the dew point. 



62 PII TSICAL ME A S UREMENT. 

Exercise 36. — To determine the dew point with Darnell's 
Hygrometer. 

Lower the temperature in the bulb in the manner just 
described, being careful not to breathe upon the bulb dur- 
ing the operation. Note the temperature £, at which dew 
begins to form on the bulb, and also the temperature t\ 
at which it disappears. The average of t and t' is the 
dew point. 



INTRODUCTION TO WORK IN ELECTRICITY 
AND MAGNETISM. 



USE OF THE MIRROR, TELESCOPE, AND SCALE. 

Ox page 3-i the theory of the optical lever is given, 
with general directions for adjusting the telescope and 
scale. It is there shown that if R cm. be the distance 
from the mirror to the scale, N cm. the difference in read- 
ing, and a the angle through which the mirror has been 

turned, — , r/r> 

tan 2 a = N/R. 

For small angles, we may write — 

i N 
tan a = -J — . 

R 

The correction for larger angles is approximately as 
follows : — 

For N/R = .5, tan a = J — - .014. 

R 

For N/R = .25, tan a = ^ — - .0019. 

R 

For N/R = .1, tan a = I — - .0002. 

R 



ELECTRICITY AND MAGNETISM. 63 

Iii angular measurements the unit angle is taken as 
57°. 296, this being the angle for which the arc equals the 
radius. The value of one scale division in degrees of arc 
for small angles is 

1 1 o7°.296 _ 28°.648 
2B~ 2 " B ~ B ' 

Here the angle is taken as proportional to N. For 
angles of any size 

a = i- the arc whose tan is N / B. 

In working with swinging magnetic needles, it is often 
found that they take a long time to come to rest. In the 
case of a magnetometer, the swings may be quickly reduced 
by presenting and withdrawing a weak magnet in such a 
way that the action between the magnet and needle, when 
the former is presented, is to oppose the motion of the 
latter. A galvanometer needle may be brought to rest by 
opening and closing a key in the circuit, in the natural 
time of the swing of the needle, the circuit being closed 
during the time the needle is swinging in the direction 
opposite to that in which the current would deflect it, and 
opened during the return swing. It is not necessary to 
wait for the needle to come completely to rest before tak- 
ing the reading, as the point of rest can be found by swings, 
as in the balance.* 

BATTERIES. 

Gravity Cell. — The most useful cell for resistance 
work in a large laboratory is the gravity Daniell, since 
it is not injured if it is left by accident short-circuited, it 
does not require to be taken apart after use. and its E.3I.F. 

* For a complete discussion of the reduction of scale readings to the 
angle, and its functions, and of damping and logarithmic decrement, see 
Kohlransch. 



64 rn vsi(\ 1 l mi\ i s rn km ent. 

is fairly constant. It cannot be used, however, where large 
currents are required, as its internal resistance is high. It 
consists, briefly, of a jar containing a copper plate at the 
bottom in a saturated solution of copper sulphate. On 
this floats a lighter solution of zinc sulphate, and in this is 
the zinc plate. It is not necessary to put any zinc sulphate 
in the jar, as it is soon formed by the action of the cell 
itself. The UJI.F. is about 1 V. 

Storage Cell. — If a constant current of some size 
(1 ampere or more) be required, the storage cell is best 
used. It consists of two sets of lead plates covered with 
a paste of oxide of lead, the alternate plates being connected 
to the + and — poles of the cell respectively. The liquid 
is a 20 per cent solution of H 2 S0 4 . For the theory of 
its action, see text-books on physics. The cell is likely to 
be injured if its current exceeds 6 amp. per square ft. of 
positive plate, hence it must not be short-circuited. The 
1J.3I.F. is about 2 V. until it is nearly discharged, when 
it begins to fall. It should not be used after the E.M.F. 
falls below 1.8 V. 

To charge the battery connect the cells in series, the 
positive terminal of the battery to the positive wire of the 
dynamo, or charging battery, with an ammeter in the cir- 
cuit. The positive plate is brown in color, and there 
are usually one more negative than positive plates in a 
cell. The charging current should be about f the maxi- 
mum discharge. The cell is charged when the reduction 
of the oxide on the negative plate has proceeded so far 
that hydrogen bubbles are given off rapidly. A storage 
battery should never be allowed to remain discharged. 
A battery which has deteriorated by being allowed to 
stand discharged for some time may often be improved by 
repeated charging and discharging. 



ELECTRICITY AND MAGNETISM. 



65 



The Chromic Acid Cell or Dip Battery. — This cell may 
be used when a storage cell is not at hand, and a strong 
current is desired. It consists of two carbon plates dipping 
in a solution composed of 1 part of chromic acid, or potas- 
sium bichromate, 2 parts of sulphuric acid, and 10 parts of 
water. The amalgamated zinc plate which slides up and 
down between the carbon plates should be raised out of 
the liquid when the cell is not in use. 
The internal resistance is small, and the 
FJf.F. is about 1.8 V. 

The LeClanche Cell. — This cell soon 
polarizes when in action, and, therefore, 
is not suited to give a continuous cur- 
rent; yet it rapidly recovers its F.3I.F. 
when at rest, and is very useful when the 
circuit is closed for but a few moments at 
a time, as in electric bell and telephone 
circuits. The electrodes are zinc and 
carbon in a solution of sal ammoniac 
(XH 4 CI). To get rid of the hydrogen 
of polarization, the carbon is surrounded 
by an oxide of manganese. Sometimes 
the carbon and the oxide of manganese 
are placed in a porous cup. The E.3LF. 
is about 1.1 V. 

Arrangement of Cells. — If we have 
six cells, they may be connected in two 
ways, in series (Fig. 23), that is, the 
copper of one to the zinc of the next, or 
in multiple arc (Fig. 21), the coppers being connected 
together and the zincs together. 

In series the F.3LF. and the internal resistance are 
both multiplied by six. In multiple arc the F.3I.F. is 




Fig. 23. 



66 



PHYSICAL MEASUREMENT. 



the same as for a single cell, but the internal resistance 
is divided by six. Then, from Ohm's law, if E is the 

E.3T.F. of one cell, r its internal re- 
+^ j- sistance, 11 the resistance of the rest 

of the circuit, and O the current, — 



(Series) C = — 



6E 



(Multiple Arc) C = 



B + 6r 
E 



fi + r/6 



If we have MN cells, consisting of 
N groups in series, each group consist- 
ing of M cells in multiple arc, the cur- 
rent will be — 

NE 



C 



E + Nr/M' 



Fig. 24. 



It may be shown that the maxi- 
mum current will be produced by 
arranging the cells so that the in- 
ternal resistance equals the external. 
When the external resistance is great 
compared to the sum of the resist- 
ance of the cells, they should be ar- 
ranged in series, thus producing the 
greatest E.M.F. 
The Commutator. — To change the direction of the 
current in any part of the circuit, the commutator shown 
in Fig. 25 may be used. It consists of six mercury cups, 
A, B, 0, D, E, F ; A and D, and B and C being connected 
by wires. Two metal rockers connected by an insulating 
bar complete the circuit. If the current comes in at D 
and goes out at 6', with the rockers as shown in Fig. 25, 
tin* current in Or (see Fig. 40) will flow in the direction 



ELECTRICITY AND MAGNETISM. 67 

of the arrow. If the rockers dip in A and B, the current 
in Gc will be reversed. 

Connections. — Wires should be connected either by 
the regular binding-screws provided for the purpose or 
with mercury cups. Their ends should be scraped so that 
they are bright, and if they are not tight in the binding- 
screws they may often be made so by doubling over the 
ends before inserting them. The mercury-cup connection 
is convenient when the circuit has to be often made and 
broken. 




Fig. 25. 

Amalgamation of Zinc. — First clean the zinc, and place 
it in dilute H 2 S0 4 until effervescence begins, then place it 
in another dish, and pour a little mercury over it. Too 
much mercury will render the zinc rotten, so only enough 
should be used to cover the surface. 

GALVANOMETERS. 

The Pointer Galvanometer. — Fig. 26 shows a common 
form of galvanometer for use in the introductory work in 
electricity. The current enters and leaves the galvanom- 
eter by the binding-posts, B, passing through the coil, (7, 



68 



PHYSICAL MEASUREMENT. 



inside which a magnetic needle hangs which is deflected 
by the current. This deflection is read by means of a 

pointer which moves over a 
circular scale divided to de- 
grees. The pointer should 
hang so as to just clear the 
scale, that the danger of 
parallax in taking the read- 
ings may be as small as 
possible. This adjustment 
may be made by raising or 
lowering the screw, to which 
the suspension fibre is at- 
tached. The pointer, before 
the work is begun, must be 
made to point to zero when 
no current is passing through 
5^ the galvanometer. If a single 
needle is used, this will point 
north and south under the 
influence of the earth's field. In this case, the whole in- 
strument must be turned till the pointer comes to zero. 
If, however, there is an astatic pair of needles, — that is, two 
needles with their poles pointed in opposite directions, one 
needle hanging in the coil, and the other above it, — they 
will be but little affected by the earth's field. It will, 
in this case, be held in position by the elastic force of 
the suspending fibre ; hence, this must be turned by means 
of the milled head until the reading is zero. This instru- 
ment is, of course, not suited to work of the highest ac- 
curacy, but is extremely convenient when this is not 
required, Most galvanometers of this pattern cannot be 
used as tangent galvanometers, except for very small de- 




Fig. 26. 



ELECTRICITY AND MAGXETISM. 



69 



flections. They are generally of low resistance, that is, 
not over two or three ohms. 

The Thomson Differential Galvanometer (Fig. 27) con- 
sists essentially of two pairs of coils, with a needle in the 
centre of each pair. These needles are borne on an alumi- 
num wire or a glass fibre of sufficient stiffness. This also 
carries the mirror and a damper 
of aluminum foil, which, by its 
resistance to the air, tends to 
bring the needle quickly to rest. 
The whole system is hung on a 
silk, or better, a quartz, fibre. 
The direction of the needles and 
mirror, and the sensibility of the 
instrument, may be varied by 
turning and by raising or lower- 
ing the control magnet, the sen- 
sibility being least when the mag- 
net is low, that is, quite near the 
upper needle in reference to the 
lower. The sensibility will be 
proportional to the square of the 
time of the swing of the needle 
(see p. 39). A time of swing of 
about six or eight seconds is con- 
venient for most purposes. If the 
needle does not seem to swing freely, the fault may usually 
be corrected by levelling the instrument by means of the 
screws. The coils are generally so connected to the bind- 
ing-posts that, when the middle two are joined by a wire, 
and then the circuit is closed by connecting to the two end 
posts, the coils are in series, and the largest deflection is 
produced. For the differential use, see p. 89. These gal- 




Fig. 27. 



PHYSICAL MEASUREMENT. 



vanometers are generally wound for high resistance 
(sometimes many thousand ohms) ; but of course low- 
resistance instruments may be arranged in the same way, 
and, in fact, some are provided with 
both high and low resistance coils. 

A rather more common form of 
low-resistance instrument is that pro- 
vided with only one pair of coils, as 
shown in the tripod galvanometer, 
Fig. 28. 

The D'Arsonval Galvanometer 
(Fig. 29) differs from those already 
described, in that the coil swings, 
while the magnet is stationary. It 
consists of a horse-shoe magnet, be- 
tween the poles of which a rectangu- 
lar coil of wire is suspended, being 
held at the top by a wire or strip of 
metal fastened to a screw, which may 
be raised or lowered, and at the bot- 
tom by another wire fastened to a 
tongue of metal provided with an 
adjusting-screw, by means of which 
the tension of the suspending wires 
may be varied. These wires also 
serve to convey the current to the 
coil. To increase the strength of 
the magnetic field, a piece of iron 
fastened to the bar supporting the coil is introduced 
inside the coil, so that the coil swings around it. The 
position of the mirror may be adjusted by means of the 
adjusting-screw. This mode of suspension makes the gal- 
vanometer very " dead beat;" that is, it comes to rest 




Fig. 28. 



ELECTRICITY AND MAGNETISM. 



71 




Fig. 29. 



almost without a swing, and can, therefore, be read very 
quickly. Of course the resistance may be high or low, 
according to the windings of the coil. 

The Ballistic Galvanometer. 
— Galvanometers provided with 
heavy needles and no dampers, 
so that the swings are slow and 
the decrement small, are called 
ballistic galvanometers. They 
are used to measure currents of 
very short duration, the quan- 
tity of electricity (the current 
times the duration) being pro- 
portional to the throw of the 
needle. 

Shunts. — If it is desirable 
to send only a part of a cur- 
rent through the galvanometer, it may be placed in a 
shunt, and the fraction of the current passing through 
will be determined by the ratio of the resistance of the 
galvanometer and that in the other arm of the shunt. By 
the law of shunts, — 
C, fit 
C 2 iV 
where is the current and R 
is the resistance. 

The Resistance-Box is a 
wooden case containing a set 
of coils, connected as shown in 
Fig. 30. The coils of German- 
silver wire, iJj, _R 2 , etc., of 
known resistance, are connected 
to the blocks of metal, B x , B^ etc. These blocks, as well 




72 PHYSICAL MEASUREMENT. 

as the plugs, P n P 2 , have a resistance so small that it 
may be neglected in comparison with i? 19 72 2 . When 
the plugs are all inserted, the resistance of the box is 
reckoned as zero. When P 1 is removed, a resistance R x 
is put into the circuit, AB. The resistance of the coils 
is usually marked upon the top of the box. 



MAGNETISM. 



ANGLE OF INCLINATION. 

The angle of inclination is the angle which the lines 
of force of the earth's field make with the horizontal. 
The dip-needle consists of a magnetic needle supported 
so that it rotates freely in a vertical plane about its 
centre of gravity. The amount of inclination is read 
from both sides of the graduated circle. The observation 
is liable to two sources of error : — 

I. The magnetic axis of the needle may not coincide 
with its axis of figure. 

II. The axis of suspension may not pass through the 
centre of gravity. 

Exercise 37. — To determine the angle of inclination of the 
earth's magnetism with the dip-needle. 

Caution. In this, as in all magnetic observations, see that 
all iron, such as keys, knives, etc., is removed from the person. 

Remove the needle, set the circle in the magnetic 
meridian with the aid of a compass, and level the instru- 
ment. Put the needle in place, and read the position of 
both ends. Call the readings i x and i 2 . To eliminate 
error II., a second pair of readings, i s and i 4 , must be 
taken with the needle inverted, which may be done, if 



MAGNETISM. 73 

the circle is arranged to rotate about a vertical axis by 
turning the circle through 180°, if it does not rotate by 
turning the needle over in its bearings. To eliminate 
error I., the magnetism of the needle must be reversed by 
stroking its poles with the like-named poles of a strong 
magnet. In order to obtain an even distribution of the 
magnetism in the needle, it is best to stroke from the mid- 
dle to one end on one side, then from the middle to the 
opposite end on the same side. Turn the needle over, 
and repeat the same operation on the other side, using, of 
course, the opposite end of the magnet for the two ends 
of the needle. This cycle of operations may be repeated 
till the desired strength is obtained. Four more readings, 
i 5 , i 6 , i 7 , i 8 , corresponding to those already taken, are now 
to be made. The average of these eight readings will be 
the correct value for i. 



INTENSITY OP THE EARTH'S FIELD. 

The lines of force of the earth's field make a certain 
angle with the horizontal, as found in the preceding exer- 
cise. The force F may then be resolved into two com- 
ponents, one vertical, T", the other horizontal, H. AVe 
shall proceed to determine the values of H, from which, 
knowing the dip, we can easily compute T" 

Exercise 38. — To determine the horizontal component, H, of 
the earth's field. 

Value of MH. — A magnet which is free to rotate 
about its centre in a horizontal plane will be drawn 
toward the plane of the magnetic meridian with a force 
which is proportional to the horizontal intensity of the 
earth's field, H, and the moment of the magnet, 31. The 



PR rSIC. 1 L M h\ 1 SUREMtiNT. 



directive force is then MIL Since the swings of the mag- 
net are harmonic, its time of oscillation is — 



V Mil ' 



whore lis the moment of inertia of the magnet, then — 

where Zand t are to he determined. 

Determine the moment of inertia I, of the deflecting 
magnet JYiS, from the formula — 

J= (ll" + 4") m ' (See p. 40.) 

where 21 equals the length, r equals the radius, and m the 
mass of the cylindrical magnet. 

Determine the time of oscillation, £, of iViS, by method 
1, p. 41, taking 20 oscillations. In order that the direc- 
tive force may be only that of the earth's field, the sus- 
pending thread must be without torsion. A fibre of 
unspun silk very nearly fulfils this condition. 

From other considerations we are able to obtain the 




Egg=h 



Fig. 31. 



value of M/IL From these two expressions, by eliminat- 
ing J/", we obtain the value if. 

Value of MjH. — MjH is determined as follows (see 



MAGNETISM. 75 

Fig. 31) : Suppose a small magnetic needle of pole strength 
// and length 2V to be held deflected through an angle, </>, 
in equilibrium, by the horizontal component of the earth's 
field, H, and a field of strength, F, due to a magnet, JV#, of 
pole strength //, and length 2Z, whose centre is at a dis- 
tance, i2, from the centre of the needle, and whose ends 
are in a line perpendicular to the plane of the magnetic 
meridian through the centre of the needle. Then the two 
forces acting on the needle, and holding it in equilibrium 
at an angle <£ with the meridian, are 2H^ I ' sin <£ = 2 Fyi! I ' 
cos </>. Therefore — 

F sin d> , , 
— ■ = . -£. = tan <£. 

H cos <£ 

If the dimensions of the needle be small with reference 
to R and ?, we may call R the distance from the centre of 
NS to either end of the needle. Then the strength of field, 
F, which is defined as the force on unit pole, will be — 



Calling the moment of the magnet 2^1 = M, we have ■ 

2MB A F 12 MR , , 

and — = = tan <f>, 



(B*-l*)*' H R(R 2 -l 2 ) 

Whence, 

M (E 2 -l 2 )\ . 
— = ^ L. tan <f>. 

H 2R * 

To determine M/R proceed as follows (see Fig. 32) : 
Remove the deflector JYS to a distance from the mag- 
netometer, and determine the zero position of the magnet- 
ometer needle by means of the mirror and scale. Place 
JYS at a v and note the deflection ?i 1 . Reverse JVS by turn- 
ing its stand through 180°, and note a second deflection n\. 



76 PHYSICAL MEASUREMENT. 

Make similar observations ra 2 , n' 2 , at 6 X . From the aver- 
age of these four observations, JVj, determine the value of 
tan 0, tan <£=iV r 1 /2t7, where <i is the distance from the 
magnetometer needle to the scale. 



a*. 5S. .^3...jb 2 



+ 



Nc=A=,S 



H, ° Ri 



l.....i.t.i.i.i.l.ni.i.i.i.i.iil.i.i.y^ . 1. 1 .l.i. i. i.i. 1. 1. 1. 1, hi. 1. 1. nt. I if 

T 



T 



F/gr. 32. 

Find the corresponding value of MJH, — 

— = ^ — i Z_ tan d)! . 

Place iVW at a 2 and £ 2 , and, proceeding as before, find 
tan <t> 2 , from which obtain a second value of Mj H, — 

— = ^ — = ^- tan <f>o . 

# 2 J? 2 ^ 

From the average of these two values of M / H, and the 
value MR obtained above, get II by eliminating M. 
Thus, — 

For the correction for torsion, see Kohlrausch, p. 222. 



ELECTRICITY. 77 



ELECTRICITY. 



DETERMINATION OF RESISTANCE. 

The current which flows through a conductor is. by 
Ohm's law. directly proportional to the electromotive force, 
and inversely proportional to the resistance. 

If we assume that our electromotive force remains con- 
stant, the resistance will be inversely proportional to the 
current. If in two cases the current, with the same elec- 
tromotive force, is found to be the same, the resistances 
also must have been equal. A simple method of deter- 
mining resistance is to measure the current with a galva- 
nometer when the given resistance is in circuit, replace the 
given resistance by a variable known resistance, which is 
adjusted till the same deflection of the galvanometer is 
obtained as befure. This is known as the method of sub- 
stitution. This method depends for its accuracy upon the 
assumption that the electromotive force remains constant 
during the determination, a condition which is seldom 
exactly met in practice. 

The apparatus required for this method is a battery of 
constant F.3I.F.* a simple galvanometer, and a resistance- 
box. For the battery, either a gravity cell or a storage 
battery may be used. 

^Exercise 39. — To determine the resistance of a conductor by 
the method of substitution. 

Connect the apparatus as shown in Fig. 33. Place 
the galvanometer so that the pointer is at 0. throw the 
unknown resistance. B 2 . into the circuit by setting the 
arm of the switch S\ upon point A. If the galvanometer 



rs 



VII TSICu 1 L MIL 1 SUllEMENT. 



is provided with more than one set of coils, use that 
set which will give a deflection nearest to 45°. If the 
deflection is too large in any case, a resistance, H, may 

be put in the main 

-'ftflRftftT ^ circuit sufficient to 

bring the deflection to 
the desired amount. 
Note the deflection 
of the galvanometer, 
and throw R x in cir- 
cuit by connecting S 
to point B, and vary 
R x till the deflection is exactly the same as before. It is 
then obvious that B 2 = R 1 . 

Caution. The circuit should never he left closed except 
while readings are being taken. 





Fig. 33. 



WHEATSTONE'S BRIDGE. 

If a current be divided as shown in Fig. 34, then, 
according to the principle of Wheatstone's bridge, if 
B 1 : li 2 :: B s : i2 4 , a 
current passing from 
A to B will not pro- 
duce a current 
through the galvano- 
meter between C and 
D ; or, more broadly 
stat cd, when the 
above condition is ful- 
filled, a difference of 
potential between A 
and B cannot produce a difference of potential between 
U and I). 




ELECTRICITY. 



79 



Wheatstone's bridge affords the most accurate known 
method of measuring resistance, since it is wholly inde- 
pendent of the changes of U.M.F. in the battery. The 




Fig. 35. 




connections for the box form of bridge are shown in 
Fig. 35. 

In the form of bridge 1 shown in Fig. 36, B s and R± 
constitute a continuous wire upon which moves a sliding 




Fig. 36. 

contact. If we assume that the wire is uniform in diame- 
ter, the resistance of its parts will be proportional to their 

1 The box form of bridge with a high-resistance galvanometer is best 
adapted for high resistances, the slide-bridge with a low-resistance galva- 
nometer for low resistances. 



80 PHYSICAL MEASUREMENT. 

lengths, which are read off from a scale upon which the 
wire rests. The error of reading will affect the result 
least when Ii\ and J{ 2 are as nearly equal as possible; i.e., 
when the contact is near the middle of the wire. 

To avoid the effect upon the galvanometer of the induc- 
tion current produced by closing the battery circuit, K x 
should always be closed before K 2 . 

It is evident that the battery circuit and the galva- 
nometer are interchangeable. 

Exercise 40. — To measure the resistance of a conductor "with 
Wheatstone's bridge. 

Connect the bridge as shown in Fig. 36. All the 
wires used in making the connections in the bridge cir- 
cuit should be so large that in ordinary measurements- 
their resistance may be neglected. Care should be taken 
that all contacts are clean and firmly made. The plugs 
should be inserted with a twisting motion to insure firm 
contact. Find by trial what resistance must be used in 
the box ijj, to bring the galvanometer to zero with the 
contact D (Fig. 36), near the centre of the wire. If the 
galvanometer is not very sensitive, move the contact till 
the needle begins to move to the right, and note the read- 
ing. Take another reading at the point where it begins 
to move to the left. Take the mean of these readings, and 
compute the resistance H 2 , from the law of the bridge. 

Exercise 41. — Resistance of a galvanometer by Thomson's 
method. 

If the galvanometer be connected in place of i? 2 , as 
shown in Fig. 37, a current will flow through it when 
K x is closed. Then, if K 2 be closed, the passage of any 
current through 6 will produce a change in the deflection 



ELECTRICITY. 



81 



of the galvanometer, as it must either increase or decrease 
the amount of flow through B 2 . If, however, the bridge 
is balanced, no current will flow through 6 ; hence we 
have only to adjust the sliding contact until no change is 
produced in the deflection of the needle by opening and 
closing K 2 . This method enables us to dispense with the 
aid of an additional galvanometer, which would be required 
in 6 by the method used in Exercise 40. 

Place the galvanometer in the resistance-arm, and close 
K x . If the deflection of the needle is too great, it may 



r 



Ri 



L 



Ra 



R 4 




Fig. 37. 



be reduced by means of a permanent magnet, or by intro- 
ducing a resistance into the battery circuit. Adjust the 
contact until the bridge is balanced. 



Exercise 42. — Resistance of a battery by Mance's method. 

The battery is placed in the resistance-arm, as in Figs. 
38 and 39. If now K 2 be closed, there will be a deflec- 
tion of the galvanometer, which will not be changed in 
amount by closing K x , so long as the bridge is balanced. 
The theory of the method is as follows : The current from 
the battery flows through B, dividing at D, where a part 



82 



PIIYSICA L ME A SUBEMENT. 



flows through 3 and 1, and a part through 6 to ( 7 , and back 
to the battery. If we also close K x , a part of the current 
will flow from B to A through 5, since A and B are not 




Fig. 38. 



at the same potential. Closing K x changes the difference 
of potential between A and B by making an additional 
path for the current, and thus changing the resistance 




between these points. By the principle of Wheatstone's 
bridge however, no change of potential between A and B, 
when the bridge is balanced, can produce a change in the 
difference of potential between C and J). 



ELECTRICITY. 83 

Connect the bridge, as in Fig. 38, and adjust for a bal- 
ance, as in the preceding exercise. If the deflection is too 
great, reduce it by means of a magnet, or by introducing 
resistance into the galvanometer circuit. 

The accuracy of this method depends upon the assump- 
tion that "the E.3I.E. of the battery remains constant. Since 
this condition is never met, the method is never strictly 
accurate, and is wholly inapplicable to batteries which 
polarize rapidly — the so-called open-circuit batteries. 

Exercise 43. — Resistance of a battery by Ohm's Method. 

Connect a battery of constant E3LF. in circuit with a 
tangent galvanometer of resistance g, and a resistance-box, 
and put in sufficient resistance r x to give a deflection d 1 , 
somewhat greater than 45°. Call the current C x . 

C x = K tan d v l 

Now insert sufficient resistance r 2 to bring the deflection 
c? 2 to about half d 1 . Call the current (7 2 . 

C 2 = JTtanrf 2 . 

Then, calling E the E.M.F:, and R the resistance of the 
batteiy, if all connecting wires are so large that their re- 
sistance may be neglected, — 

C,= * , C 2 = 



R + r x + g ' " B + r 2 + g' 

C x _ tan d x _ R -f- r 2 -f g u 
C 2 tan d 2 R + ?\ + g 
R = tan d 2 (r 2 + g) — tan cl x (r\ + g) ^ 
tan d ± — tan d 2 

The circuit should be closed only long enough at a 
time to take a reading, so as to avoid a change in E.M.F. 
of the battery from polarization. 

1 See, p. 91. 



84 



PR YSICA L MEASUREMENT. 



Four readings f each deflection should be taken, one 
at each end of the pointer attached to the needle, and a 
second pair after the current has been reversed. The con- 
nections of the reversing-switch are shown in Fig. 40. 



SPECIFIC RESISTANCE. 



The resistance of a wire is inversely proportional to its 
cross-section, and directly proportional to its length and to 

a factor which depends 

r\\ — 



By 



inn 



R 



upon the nature of the 
substance. This factor, 
#, is known as the specific 
resistance, and may be 
defined as the resistance 
of unit length and unit 
cross-section of the sub- 
stance. 

Determination of the 
Specific Resistance by- 
Fall of Potential. — This 
method can be used to 
advantage Avhen the re- 
sistance to be measured is 
small. The resistance, R 
(Fig. 41), to be mea- 
sured is connected in circuit with a battery, i?, a control 
resistance-box, R±, and a standard resistance, r, conveni- 
ently one ohm. A galvanometer circuit is introduced as a 
shunt, first between the terminals of i2, and then between 
the terminals of r. The current flowing through the gal- 
vanometer will be, according to the law of shunts, propor- 
tional to the resistances, R and r. For, if C and C" be 
the currents passing through R and r in the two cases, 




MD 2 -^ 



Fig 40. 



ELECTRICITY. 



85 



and C\ and C 2 the corresponding currents through the 
galvanometer, we have — 

£. = *, and ^ = _ZL. 
C g[ C" g 

If the resistance, ^, of the galvanometer is so large that 
the current flowing through it is negligible in comparison 
with the current in the main circuit, — 

C = C", and C Y /C, = R/r. 

In the mirror galvanometer the current may be consid- 




Fig. 41. 



ered, for small deflections, proportional to the deflection. 
The amount of the deflection may be regulated by changing 
the resistance of the main circuit by means of R x . 

Exercise 44. — Determine the specific resistance of German 
silver. 

Choose a wire of such length that its resistance is not 
very far from one ohm. The ends are to be soldered to 
pieces of large copper wire, which dip in the mercury cups, 



86 PHYSICAL MEASUREMENT. 

0j, <\ 2 (Fig. 41). The standard ohm is connected to cups 
e? 8 , e 4 . Adjust i2j so that the larger deflection of the 
galvanometer shall be not more than 20 cm. on the scale. 
Reverse the current in the galvanometer for each reading. 
Find A\ as indicated above. Measure the length Z, and 
cross-section a, of the wire. Then — 

S = Ea/l. 

Temperature Coefficient of a Metallic Resistance. — The 

resistance of metals increases with the temperature. Some 
alloys are exceptions to this law, and alloys may be made 
of such proportions that within certain limits their resist- 
ance does not perceptibly vary with the temperature. 

The resistance, R n at any temperature, £, of a metal 
may be expressed by the formula, — 

(A) E t = B (1 + at), or more exactly, 

(B) R t = £ (l + at + fit*), 

where R is the resistance at 0°, and a and /? are coeffi- 
cients to be determined by experiment. Equation (A) is 
sufficiently accurate for most purposes. From (A) we 

have — 

a = (E t — Ji )/E t. 

Exercise 45. — To determine the temperature coefficient of 
copper. 

The wire to be tested is wound on a small wooden 
block and placed in a test-tube, through the cork of which 
passes a thermometer. The test-tube is filled with petro- 
leum, and immersed in a water-bath. The bath is first 
filled with melting ice; and, when the temperature of the 
oil is 0°, the resistance of the wire is determined by the 
Wheatstone-bridge method. The bath is then filled with 
water, which is brought to 100° over a Bunsen burner, 



ELECTRICITY 



87 



and another determination made. Three determinations 
should be made at each temperature. 

RESISTANCE OP AN ELECTROLYTE. 

The methods of measuring resistance already described 
are not adapted to electrolytes, because of the apparent 
change of resistance due to polarization when a steady 
current is employed. This difficulty may be overcome by 
employing an alternating current, which does not flow in 
one direction long enough for the polarization to become 
noticeable. As the ordinary galvanometer does not respond 
to an alternating current, an electro-dynamometer, or more 




Fig. 42 a. 



Fig. 42 b. 



simply a telephone, is inserted in the Wheatstone's bridge 
in place of the galvanometer. To obtain the alternating 
current a small induction coil, having a low-resistance 
secondary circuit, is used, the secondary terminals of 
which are connected in place of the battery wires in the 
Wheatstone's bridge. 

The electrolyte whose resistance is to be determined 
is placed in the cell shown in Fig. 42 <x, if of low resist- 
ance, or the one shown in Fig. 42 5, if of high resistance. 
The electrodes are of platinum which have been coated 
with platinum black. The Kohlrausch form of bridge 



88 



Vll YSICAL ILEA S UREMENT. 



shown in Fig. 43 is well adapted for this method. The 
bridge wire is wound upon a movable cylinder. Contact 
is made with this wire by means of a metallic wheel at D, 
which is held against the wire by springs. 




Fig. 43, 



Exercise 46. — Resistance of a battery with the alternating 
current. 

In this case no special cell or electrodes are required. 
Make the connections as indicated above, and vary the 
point of the contact till the telephone becomes silent. It 
is well to place the induction coil in an adjoining room, to 



ELECTRICITY. 89 

avoid disturbance from its sound, as well as from possible 
direct inductive effect upon the telephone. If the tele- 
phone is very sensitive, absolute silence may not be ob- 
tained. In this case the student must judge when the 
sound has reached a minimum. 

The battery should be disconnected except when the 
measurements are being taken. 

Exercise 47. — Specific conductivity o- an electrolyte. 

Specific conductivity is the reciprocal of specific resist- 
ance. The resistance of the celli? 1 . when filled with a 
liquid of known specific conductivity jfi^, is first deter- 
mined. Then if the resistance of the cell when filled 
with the given electrolyte be i?. 2 . the conductivity K. 2 , of 
the electrolyte is — 



K, = 



B 9 



The resistance constant, y (or resistance capacity), of 
the cell is K x i? 1 . and the specific conductivity of any 

substance which has resistance E in the cell is — 

R 

Caution. On account of t~ ' : ting effects of the current, 
the circuit should be kept closed the shortest time possible. For 
the some reason the current should not be large, especially if the 

cell is small. In determining the resistance the temperature 
should be noted for each observation. 

MEASUREMENT OF RESISTANCE BY THE DIFFERENTIAL 
GALVANOMETER. 

If two equal coils of a galvanometer are similarly 
situated with respect to the needle, the needle will remain 
at rest when equal currents are flowing in opposite direo- 



90 



PHYSICAL MEASUREMENT. 



(ions through the coils. If the connections are made as 
shown in Fig. 44, the resistances R x and jR 2 will be equal 
when the needle is at rest. The balance of the instrument 
may he tested by placing two known equal resistances in 
the circuit, and noting whether any deflection occurs. By 
inserting a commutator in the circuit (see Fig. 45), the 




( ^ 






















f 


> 




t — 




R 2 


—^ 



Fig. 44. 



Fig. 45. 



necessity for a perfect balance of the instrument is 
obviated; for if a resistance, H 2 , gives no deflection be- 
fore reversing the current, and a slightly different re- 
sistance, II' 2 , is required to produce a balance after 

reversal, — 



CALIBRATION OF GALVANOMETERS. 



91 



CALIBRATION OF GALVANOMETERS. 



THE TAXGEXT G AL YAXOMETER . 

The simplest form of galvanometer consists of a large 
circle of wire with a small needle supported at its centre. 
(See Fig. 46.) 

If the circle be placed in the plane of the magnetic 
meridian, and a current be sent through it, the needle 
will be deflected through an angle 
whose tangent is proportional to the 
current. For the magnetic force 
due to the current will be, as is 
shown on p. 75, proportional to the 
tangent of the angle of deflection. 
If be the current and a the de- 
flection, we may write, — 

C — K tan a, 

where K is a constant depending 

on the radius of the circle, r, the 

number of turns, n, and the horizontal intensity of the 

earth's field, H. 

In absolute units the unit of current is defined as that 
current which, flowing through a length of 1 cm., of an 
arc of 1 cm. radius, exerts a force of 1 dyne upon a unit 
magnetic pole at the centre. If the unit current flows 
through an arc of length I, and radius r, the force upon 
the unit pole at the centre is obviously l/r 2 . 

A current of strength O will exert upon a pole of 
strength ^, a force — r j / a 

The force of a coil of n turns will be — 

„ 2 tzu 




Fig. 46. 



92 PHYSICAL MEASUREMENT, 

The force of the earth's field on a pole of strength ^ is 
7//x. Then, if we call the force of unit current upon unit 
pole — 

2 wn J r = G, 
we have — 

— '£ = tan a ; C = — tan a, 
i//* ' G ' 

in absolute units, or, in amperes, calling 10 H '/ G= K, — 

C = K tan a. 

It is evident that K may be computed for any place where 
II is known. 

An experimental method of determining K consists 
of measuring the current which corresponds to any deflec- 
tion of the galvanometer, by weighing the products of 
electrolysis. 

Exercise 48. — To calibrate a tangent galvanometer. 

Prepare an electrolytic cell containing 100 parts by 
weight of water, 20 parts of copper sulphate, and a few 
drops of sulphuric acid. The electrodes may be two thin 
plates of copper as large as can be conveniently used in the 
vessel. Sandpaper both plates till they are bright. Place 
them in the solution, and connect in circuit with a tangent 
galvanometer provided with a reversing-key, a resistance- 
box, and a storage-cell or two chromic acid cells. Adjust 
the resistance for a deflection of about 45°. Remove the 
negative plate, rinse in water, then in alcohol, dry over a 
Bunsen burner, and weigh, taking care not to touch the 
electrode with the lingers. If it is necessary to lay the 
electrode down, lay it upon clean filter paper. Now re- 
place the electrode in the solution, connect the battery, 



CALIBRATION OF GALVANOMETERS. 93 

and run for half an hour, noting the deflection every two 
minutes. Read both ends of the needle, and reverse the 
current through the galvanometer at each observation. 
Again remove, rinse, dry, and weigh the electrode as 
before. 

One ampere deposits .00828 grams of copper per second 
upon the negative electrode. The current which causes 
the deposition of m grains in t seconds will then be — 

C = 



.000328 • t 
If the average deflection during the time be a, then — 

K = amp. 

tan a 

Caution. The current must not be allowed to exceed what 
is called the critical density, i.e., it must not be larger than can 
be carried by the copper alone for the size of the electrodes 
employed. When this current is exceeded \ a hydride of copper 
is formed, ivhich may be recognized by the dark color of the 
metal deposited. 

The gain of the negative electrode is used rather than 
the loss of the positive electrode, since the latter is af- 
fected by some slight irregularities which make its use 
unreliable. 

K should also be calculated from the dimensions of 
the galvanometer, and the value of H, and the two values 
compared. 

CALIBRATION OF OTHER THAN TANGENT 
GALVANOMETERS. 

If the size of the needle is not small compared with 
the size of the coil, the needle will be acted upon by a 
field of different strength for different deflections, hence 
the current will not be proportional to the tangent of the 
angle of deflection. 



04 



PHYSICAL MEASUREMENT. 



The most practicable way to calibrate galvanometers 
of this kind is as follows : — 

Exercise 49. — To calibrate a galvanometer. 

Connect the galvanometer in series with a tangent 
galvanometer of known constant, 
or a Thomson balance, a resistance- 
box, and battery. 

(«) In case of a galvanometer 
having a pointer and circle, find the 
amomit of current corresponding to 
deflections of 5°, 10°, 15°, etc., to 
60°, and plot a curve, using deflec- 
tions as orclinates, and the corre- 
sponding currents as abscissas. The 
^-^ current corresponding to any de- 

\^J \ J flection may then be taken directly 

from the curve. 

(5) In the case of a mirror gal- 
vanometer, the galvanometer (whose 
resistance is known) is connected 
in shunt with a resistance-box i2, 
as in Fig. 47. With no resistance 
in R adjust R 1 till the tangent gal- 
vanometer gives a deflection of 
about 45°, then adjust the resis- 
tance R to obtain the deflections of 
5, 10, 15, etc., scale divisions on the mirror galvanometer, 
taking the corresponding readings on the tangent galva- 
nometer. From these data calculate from the law of 
shunts the current corresponding to each deflection, and 
plot a curve as in (a). 




Fig. 47. 



ELECTROMOTIVE FOBCE. 95 

ELECTROMOTIVE FORCE. 

Exercise 50. — Determination of electromotive force by the 
method of comparison. 

Join the cell to be tested in series with a gravity-cell 
whose electromotive force is known, a resistance-box, and 
a galvanometer. Introduce sufficient resistance to pro- 
duce a deflection between 35° and 50°. 

Reverse the connections of the gravity-cell so that 
it will send a current in opposition to the other cell, 
and note the deflection. Denote the tangent of the first 
deflection by C x , and that of the second by (7 2 . Then if 
E x be the E.3I.F. of the gravity-cell, and E 2 that of the 
cell to be tested, — 

Ci = E, + E x ^ or ^ = Ci+ C, ^ 

6-2 -t>2 — -t<\ C 1 — C 2 

Exercise 51. — Determination of the electromotive force of 
a cell by Ohm's method. 

Connect the cell in series with a resistance-box and 
tangent galvanometer whose constant K, is known. 

Introduce enough resistance to produce a deflection of 
about 50°. Call this resistance B 1 . Note the deflection 
and calculate the current 1 , in amperes. Then increase the 
resistance till the deflection is about half what it was before. 
Call this resistance i? 2 , and calculate the current 2 , then 
if b and g are the resistances of the cell and galvanometer, — 

r E r E 

W — - — ; ; — 7^ ? ^2 — 



b + g + B x ' ' b + g + B 2 ' 

jp ^1^2 (-fig — E.o) 

wo C/J 

This method is only suitable for constant cells. 



90 



PHYSICAL MEASUREMENT. 



THE QTTADRAOT ELECTROMETER. 

A discussion of the quadrant electrometer may be 
found in any text-book of physics. Briefly stated, its 
principle is as follows : Suppose the needle, n (Fig. 48), 
to be charged positively, and quadrants II, III, connected 
to the ground. Then if a positive charge be given to I, 




IV, the needle will be deflected in the direction shown by 
the arrow by a couple which for small deflections is pro- 
portional to the difference of potential between the two 
of quadrants. 
A convenient form for the laboratory is that of Edel- 
inann, which differs from the ordinary form in having the 



THE QUADRANT ELECTROMETER. 



97 



i 



o 



quadrants composed of four vertical sections of a cylinder, 
the needle being bent downwards at the ends, as shown 
in Fig. 49. The needle is suspended by a 
quartz fibre, or bifilar suspension, and the de- 
flections are read by means of a mirror and 
scale. Connec- 
tion to the 
needle is made 
through the cup 
of sulphuric 
acid c7, which 
also serves to 
remove damp- 
ness from the 
instrument. 

Fig. 49. 




CZZ3 



Exercise 52. — To measure the E.M.F. of a battery with the 
quadrant electrometer. 

Charge the needle by connecting it to one terminal of 
the water battery described on p. 144, the other terminal 
of the battery being connected to earth through the water 
or gas pipes. Connect one pair of quadrants permanently 
to earth through the mercury-cup B. Bring the other 
pair to zero potential by connecting A and B for an 
instant with a wire. Xow note the zero reading, connect 
the terminals of the standard cell to A and B, and note the 
reading on the scale. Disconnect the cell, discharge the 
quadrants, and again connect the cell, reversing the con- 
nections so as to produce a deflection in the opposite 
direction. If the two deflections are not nearly equal, 
they should be made so by adjusting the level of the 
instrument, or turning the head which supports the needle. 
Call this double deflection S. In the same way find the 



08 PHYSICAL MEASUREMENT. 

deflection, §\ produced by the cell to be measured. Then, 
if E be the E.M.F. of the standard cell, and U' the 
E.M.F. of the cell to be measured, E/E 1 = 8/8'. 
A V hence — 

8 

A series of at least three such determinations should 
be made, and the average taken. In case no water battery 
is at hand, the needle may be charged by touching a rubbed 
glass rod to the wire leading to the needle, first attaching 
a small metal ball to the end of the wire to prevent dis- 
charge. As in this case the needle will gradually lose its 
charge, two pairs of readings must be taken with the 
standard cell, one immediately before and one immedi- 
ately after the readings with the cell to be measured. 
The average of these two deflections should be taken for 
comparison with the deflection of the unknown cell. 

Caution J. Never on any account short-circuit the cells. 
To do so ivill cause a change in E.M.F., and vitiate the measure- 
ment^ and probably ruin the standard cell. 

Caution II. See that all ivires are carefully insulated. 

Caution III. As the electrometer is sensitive to outside dis- 
turbances^ such as may be produced by charges present on the 
clothing, no one should approach the instrument while a reading 
is being taken. If these precatttions are neglected, no determina- 
tions of any value can be made. 



SOUND. 



99 



SOUND. 



VELOCITY OF SOUND. 

Exercise 53. — Velocity of Sound in solids by Kundt's 
method. 

The apparatus (see Fig. 50) consists of a large glass 
tube T, about 3 cm. in diameter and two metres long, 
closed at one end by a cork. The tube, which has been 
thoroughly cleaned, contains a little lycopoclium, or fine 
cork-dust, distributed evenly through its length. A rod 



f~^m mm u nOla 1 1 i Qj . 



Fig. 50. 



of the substance to be tested is clamped at its centre (7, 
and bears upon one end P, which projects into the tube, a 
disk of cardboard or cork whose diameter is a little less 
than the inner diameter of the tube. 

Rub the end R of the rod with a rosined leather or 
cloth. The longitudinal vibrations of the rod will produce 
stationary waves in the air in the tube, which will cause 
the dust to arrange itself in wave-like figures. These 
figures may be rendered more distinct by turning the tube 
so as to bring the line of dust a little to one side. It will 
then fall down at the points of greatest disturbance. The 
tube should be moved along till that length is obtained 
which will make the figures as distinct as possible. The 
distance between two of the points of greatest disturbance 
will be half a wave-length. Call this length ?, and the 



100 PHYSICAL MEASUREMENT. 

length of the rod L. From the theory of longitudinal vi- 
brations in rods, L is one-half a wave-length of the funda- 
mental tone emitted by the rod. The velocity of sound 
in air at 0° is 331 metres per second. Its velocity at any 
other temperature t, may be taken as 331 VI + .004 t * 
metres per second. Then, as the velocity is proportional 
to the wave-length, the velocity of sound in the rod is — 

V= 331 Vl + .004* — 

b 

metres per second. From this value of l^may be calcu- 
lated the modulus of elasticity of the rod. For — 

where E is the elasticity, and D the density, of the sub- 
stance ; Whence — 

U= V 2 D. 

To reduce this to the dimensions usually employed to 
express J?, we must divide by 9810. 

^Exercise 54. — -To determine the velocity of sound in air with 
Konig's apparatus. 

The apparatus (see Fig. 51) consists of a tuning-fork 
i% and resonator R. To the resonator is attached a tube 
which divides at a, allowing the sound to proceed by two 
paths through the arms c and d to b. The length of d may 
be varied at will. If c and d differ by half a wave length, 
the two waves will arrive at b in opposite phase and will 
interfere. This condition may be shown by the mano- 
metric flame, m, and revolving mirror, M. 

First adjust the resonator to resonance with the fork, 

* See Kohlrausch's Praktische Physik. 



SOUND. 



101 



by sliding the tube in and out till the volume of tone is 
a maximum, then attach the tube, a, bow the fork, set the 
mirror rotating, and observe the reflection of the flame in 
the mirror. It will in general present a wavy appearance. 
Adjust the length of d till this wavy appearance disappears 
and note on the scale s, the amount which d has been 
drawn out. This position of no vibration is somewhat 
difficult to determine. It can best be found by noticing 
the first appearance of waviness as the tube is drawn out, 



/^\ 



Set 



z3 



M *—m 



Fig. 51. 



and again as the tube is pushed in, and taking the mean 
of the readings. The length ?, to which the tube is drawn 
out is i of a wave length A. If the number of whole 
vibrations of the fork be w, the velocity of sound at the 
temperature of the room is n A meters per second. 



Exercise 55. — Measurement of the pitch of a tuning-fork 
with the siren. 

For a description of the siren and the principle of its 
action, the student is referred to text-books on physics. 



102 PHYSICAL MEASUREMENT. 

The fork is bowed, and the siren started with the 
registering apparatus disconnected, and brought into uni- 
son with the fork by forcing the air until the desired note 
is reached. This may be maintained either by means of 
the adjustable weight usually found on the bellows, or by 
the friction of a straw lightly pressed against the revolving 
plate. Since the quality of the sound of the fork differs 
from that of the siren, it is better to trust to beats for 
detecting unison, rather than to the musical accuracy of 
the ear. The registering apparatus is next thrown into 
gear, and allowed to run for exactly one minute, the siren 
being kept at the same pitch during that time. If the 
number of turns registered is N, and the number of holes 
in the plate ?i, the number of vibrations per second — 

Five determinations should be made. 



LIGHT. 



PHOTOMETRY 

The general method of photometry consists of finding 
a point between the source of light whose intensity is to be 
measured, and a standard source, where the light from the 
two is equal in intensity, measuring the relative distances 
of the two, and from the law of inverse squares computing 
the relative intensity. 

The commonest form of apparatus is Bunsen's photom- 
eter, where the equality of illumination is determined by 
the simultaneous observation of both sides of a paper 
screen, g, having a grease spot upon it, by means of two 



LIGHT. 103 

mirrors, M, M 1 ', Fig. 52. The support which carries the 
mirrors and screen slides along a graduated bar, on 
which the distance of the screen from the lights may 
be read. 

A second form is the photometer of Joly, in which the 
spot and mirrors are replaced by two small blocks of p'araf- 
fine, with a thin opaque screen between them. The blocks 
should be cut from a piece of paraffine of uniform trans- 
parency, and may be 5 mm. wide, 2 cm. high, and 3 cm. 
from front to back. When the lights are of equal intensity 
the two blocks will appear equally illuminated. 

The standard of light is a sperm candle burning 7.78 
grams per hour, as measured upon a specially arranged 
candle balance. For practical use the standard light is 
ordinarily a gas-flame placed behind a slit in a large screen. 





hi/* 


\H 




s 


i ^ 


_-^N 


I 


\J 




i 


V 




! s ! 

1 i 






® l 


i 

i e' 





Fig. 52. 

The slit is of such size that only the light from the central 
portion of the flame passes through it. Here the slit is 
taken as the source of light in measuring its distance. 
This standard light must first be calibrated by comparison 
with a standard candle. 

Exercise 56. — To measure the horizontal candle-power of an 
incandescent lamp with the Bunsen or Joly photometer. 

Practise making settings of the photometer until a 
number of successive readings can be taken without a 
variation of more than 3 or 4 per cent. Take a series of 



104 



PUY8ICA L M K. ISU E EM KNT. 



six readings, rotating the lamp 30° about the vertical axis 
a fur each observation. At least six settings should be 
taken for each of the six positions. Compute the candle- 
power for each position, and plot a curve showing the dis- 
tribution of light for the different positions of the filament. 



LENSES. 

Radius of Curvature of a Lens or Mirror. — The 

method of measuring the radius of curvature of large 
lenses with the spherometer is described on p. 12. The 
following method is adapted to lenses or mirrors of any 
size. A telescope, T, Fig. 53, is placed midway between 



9i<X 



U (1) 




*p- 



Fig. 53. 



the two small gas-jets, g v ^ 2 , which are a distance L 
from each other. At a distance A, about 3 metres from 
the line g x # 2 , is placed the lens, upon the surface of 
which is fastened a small scale, ss v upon which is read the 
apparent distance, Z, of the images of the two gas-jets. 
If a second pair of images reflected from the back of the 
lens is seen, they may be distinguished from the ones to be 
used from the fact that one pair will be upright, and the 
other inverted. 



LIGHT. 105 

The radius of curvature of a convex lens is — 
P 2 Al 



L -21' 

and for a concave lens, — 

2 41 

L + 2 1 

For, if a be the distance of the image g z g± behind the 
surface of the lens, and b its length, then — 

a = A + ~B ' a = 2 A + B 

/r{ \ b a 7 La 

(C) - = -, <** = _; 

(D) - = —4 — , or I = — ^_ = Xa , from (C). 
K J b A + a A + a A + a K J 

From (B), — 

(E) I = - LE . whence, (F) B = 2 Al . 
K J 2A + B J V ; L -21 

The proof for a concave lens is exactly similar to 
this. 

Focal Distance of a Lens. — The principal focus of a 
lens is the point of convergence of rays which were par- 
allel before passing through the lens. The focal distance 
of a lens is the distance between the optical centre and 
the principal focus. 

Exercise 57. — To measure the focal distance of a convex lens. 

(a) Find the relative position of the lens and a white 
screen of paper or ground glass for which a clear image of 
a distant object, as a cloud or spire, is obtained, and measure 
the distance between the image and the optical centre of 
the lens. This is the focal distance of the lens. 

(J) Second method. Place a candle or other bright 



106 



PHYSICAL M EA S UHEMENT. 



object, <\ Fig. 54, near a screen, s, and at one edge of it. 
Hold behind the lens a plane mirror, and move lens and 
mirror together parallel to the screen until a sharp image 
of the candle appears on the screen. The distance of the 
image from the centre of the lens is the focal distance of 
the lens ; for, if the rays of light from the object had passed 
but once through the lens, they would have converged at a 
point double the focal length from the lens, on the other 
side of it 




Fig. 54. 



(<?) Third method. Place the object at any convenient 
distance from the lens, and move the screen until a sharp 
image is found. Mark the position of the lens, and, without 
moving screen or object, move the lens till a second position 
is found, where a sharp image is obtained. If a be the 
distance between the object and image, and d the distance 
the lens is moved, then, — 

„ _ a 2 — d 2 
^ ~~~ 4 a 
For, since the object and image are respectively 1/2 
(a + d) and 1/2 (a — d) from the lens in the first case, 
and 1/2 (a — d) and 1/2 (a -f- d) in the second, and since 
1 //= 1 / p -\-l / q = where p and q are the distances of 
object and image from the lens, it follows that — 
1 _ 2 2 _ 4a 

f a + d a — d a 2 — d 2 



LIGHT. 107 

Exercise 58. — To measure the focal distance of a concave 
lens. 

Fit the lens in an opening through a screen, and let the 
sun's rays fall perpendicularly upon it. Behind the screen, 
on a second screen, parallel to and at a convenient distance, 
« , from it, a circular spot of light will be seen. If the diam- 
eter of the lens be <i, and the diameter of the spot be d\ the 
focal distance is, — 

/, _ ad 
J ~ d' -d 

Note. — The measurements in this and preceding exercises are to be 
taken from the optical centre of the lens. For a double convex lens or 
double concave lens of equal curvature on both sides, the optical centre 
is at the centre of the lens ; for plano-convex and plano-concave lenses, 
it is upon the curved surface. See Carhart, University Physics, Part ii., 
art. 20. For other cases and for systems of lenses, see Kohlrausch, 7th 
edition, art. 44. 

Magnifying Power of Optical Instruments. — The mag- 
nifying power of a telescope is the ratio between the angle 
subtended by the image of a distant object when seen 
through the telescope, and the angle subtended by the 
same object when seen by the naked eye. 

Exercise 59. — To determine the magnifying power (a) of a 
telescope (5) of a microscope. 

Focus the telescope upon a large scale or other object 
showing distinctly marked equal divisions, as a brick wall. 
The scale should be at as great a distance as is convenient 
from the telescope. Place one eye at the telescope, and 
look at the object at the same time with the telescope and 
with the naked eye. With a little practice the magnified 
image may be made to appear to be overlapping the image 
seen with the naked eye. Count the number of divisions 
on the naked-eye image that just cover one division on the 



108 



PHYSICAL MEASUREMENT. 



magnified image, estimating to tenths of a division. This 
number will be the magnifying power of the telescope. 

Note. — If, owing to near- or far-sightedness, the observer wears 
glasses, the glasses should not be removed for making observations with 
the telescope or microscope.. 

The magnifying power of a microscope may be deter- 
mined by a method similar to that employed in (a). The 
microscope is focused on a slide, upon which a scale is 
ruled to hundredths of a mm., and a millimeter scale is 
placed upon the stand of the instrument for comparison. 

THE SPECTROMETER. 

The spectrometer (see Fig. 55) is an instrument de- 
signed for measuring the angle of deviation of a ray of 
light. The ray of light 
is received through a 
collimator, (7, which con- 
sists of a tube provided 
with a slit and lens, and 
fixed to point exactly 
toward the centre of a 
graduated circle, upon 
which rotates the ob- 
serving telescope T, 
which also points to the 
centre of the circle. 
The telescope may be 
clamped in any position 
on the circle, and ac- 
curately adjusted by 
means of a tangent screw S. Its exact position on the 
scale is read by means of two verniers, which are often 
provided with lenses for more exact reading. 




Fig. 55. 



LIGHT. 109 

The Circular Vernier — The vernier employed in read- 
ing angular divisions is identical in principle with the 
straight vernier already described. Since there is no uni- 
form practice in regard to the number of divisions employed 
in circular verniers, the student should first notice into how 
many divisions the degree is divided on the circle, and 
then find how many divisions on the circle correspond to 
the total number of divisions on the vernier. It will then 
be easy to determine from the principal of the vernier the 
value of its divisions in fractions of a degree or in seconds. 
A common form has the degrees on the circle divided to 
fourths, and 45 divisions on the vernier corresponding 
to 44 on the circle. One division of the vernier is thus 
equal to \ of ^ or t \q of a degree, or 20 seconds. Most 
exact instruments are provided with two verniers 180° 
apart. The true reading, when the verniers read V and 
y i s 

y_ V + (P- 180°) 

2 

Exercise 60. — To measure the angles of a prism -with the 
spectrometer. 

Adjustment of Telescope and Collimator. — First focus 
the telescope for parallel rays, by observing a distant 
object or an object placed at the principal focus of a 
lens in front of jhe telescope. The adjustment is accu- 
rately made when the cross-hairs of the eyepiece show no 
parallax ; that is, when they do not appear to move with 
reference to the object as the eye is moved before the 
eyepiece. Place the telescope in line with the collimator, 
and put a source of light before the slit. The image of 
the slit should be a little wider than the cross-hair. Now 
focus the collimator till there is no parallax between the 
cross-hairs and the image of the slit. The rays entering 



110 PHYSICAL MEASUREMENT. 

the telescope are now parallel, and these adjustments 
should not be disturbed during the experiment. It is 
presumed that the telescope, collimator, and circle are 
properly levelled. Should this not be the case, the adjust- 
ments may be made by means of a spirit-level. 

Adjustment of the Prism. — To level the prism, turn 
the slit so that it will be horizontal, place the prism as 
shown in Fig. 55, with its edge at the centre of the table 
and directed towards the collimator. Now view the image 
of the slit as reflected from the prism, and adjust the level 
of the prism till the image of the slit coincides with the 
horizontal cross-hair. Turn the telescope till the image 
is seen reflected from the other face, and again level till 
slit and cross-hair coincide, then turn back to see that 
the first level has not been disturbed- 
Measurement of the Angle of the Prism. — With the 
prism in the position just described, and the slit vertical, 
set the telescope so that the image of the slit coincides 
with the vertical cross-hair, and read both verniers. In 
the same w^ay read the verniers when the image coincides 
with the cross-hairs as seen from the other side. If the 
readings be a x and a 2 , the angle of the prism is — 



A 



a-! — a 2 



2 

In the same manner measure the angles B and C. Then 
A + B+ C' = 180°. 

Exercize 61. — To determine the index of refraction of a 
prism. 

The index of refraction of a substance for air is the 
ratio of the velocity of light in air to its velocity in the 
given substance. If i (see Fig. 56) be the angle which 
the incident ray makes with the normal to the bounding 



LIGHT. 



Ill 



surface while the ray is in air, r the angle made by the 
refracted ray, and D the total de- 
viation of the ray, then — 

D = (i - r) + (•' - i0 

but r + r r = A, 
hence, for the position of minimum 
deviation where it may be proved 
that i = V and r = r\ D = 2 i — J., 
where A = 2r. The index of refraction n, is 




Fig. 56, 



n = 



sin i __ sin i (Z> + A) 



sm r 



sin i A 



?L 



To measure i), illuminate the slit with sodium light 
by placing before it a Bunsen burner, in which a platinum 
spoon is placed containing a little sodium 
salt. Point the telescope at the slit, and 
note the reading of the verniers a 1 . Then 
place the prism on the table as shown in 
Fig. 57, and move the telescope until the 
refracted image coincides with the cross- 
hair, and slowly move both 
prism and telescope till, in 
whichever direction the prism 
is moved, the deviation will 
be increased. This will be 
the position of minimum de- 
viation ; and the difference 
between this reading of the 
circle, a 2 , and the reading ai , 
is D of the formula. 




Fig 57. 



112 PHYSICAL MEASUREMENT. 

SPECTRUM ANALYSIS. 

The spectra obtained by allowing a ray of light to 
pass through a prism are of three sorts: — 

I. The continuous spectrum given by a body in the 
solid or liquid form heated to incandescence. It shows 
the colors of the rainbow in regular order, from red to 
violet. 

II. The bright line spectrum, given by bodies heated 
to incandescence in the form of vapor. Here every ele- 
ment emits light of one or more definite wave-lengths, 
thus producing bright lines which have corresponding 
definite positions in the spectrum. 

III. The dark line spectrum, obtained by viewing a 
body which gives a continuous spectrum through a vapor 
at a lower temperature, the vapor absorbing the vibrations 
of the same wave-length that it is capable of emitting, 
and thus producing dark lines at the same positions as the 
bright lines seen in the spectrum of the same vapor when 
heated to a higher temperature. 

Spectrum analysis is a method of detecting the pres- 
ence of the various elements of a compound or mixture, 
by identifying their characteristic lines in the spectrum. 

For this purpose the spectrometer may be used. As 
the amount of deviation, however, depends upon the 
nature of the glass of which the prism is composed, it 
will be necessary to calibrate the instrument. This may 
be done by observing the deviation for the principal dark 
lines of the solar spectrum (Fraunhofer's lines), and plot- 
ting a curve of wave-lengths, using the wave-lengths as 
ord i nates, and the deviations as abscissas. The wave- 
lengths may be found in Table 33. 

Instead of the spectrometer, the spectroscope (See Fig. 
58) may be used. In the spectroscope the collimator and 



SPECTRUM ANALYSIS. 



113 



telescope are fixed, or nearly so, the prism is set once for 
all in the position of minimum deviation, and the circle 
is replaced by a third tube containing a scale, the image 
of which is reflected from the surface of the prism so as 



9 L 



and slightly 
usually cor- 
Bunsen and 
at 50. Be- 
calibrated 



to appear in the telescope just above 
overlapping the spectrum. The scale 
responds approximately to that of 
Kirchoff, the sodium (IX) line being 
fore use, however, the scale must be 
by comparison with the Fraun 
hofer lines. For this purpose a 
ray of sunlight is brought 
into the room in a hori- 
zontal direction by means 
of a heliostat, ,and the 
collimator placed in a line 
with the ray. If the ^ 
light is too intense, the L 
slit should be narrowed 
till the lines are distinctly 
seen. Adjust the scale till the divisions are seen clearly, 
and the dark line (D) in the yellow is at 50. By com- 




Fig. 58. 



A a B C 



Red. 



Yellow. 



E b 



Gree>~. 
Fig. 59. 



H H' 



Blue. 



Violet. 



parison with Fig. 59 identify the principal Fraunhofer 
lines, and plot a curve, using wave-lengths as ordinates, 
and scale-divisions as abscissas. 



114 



PHYSICAL M EA S U REM EX T. 



Exerei8e (>2. — Analysis of metallic salts. 

Place the substance to be investigated in a platinum 
spoon, or form a bead of the salt on a platinum wire by 
heating the wire red-hot, and dipping it in the salt. Now 
support the spoon or wire in a Bunsen flame in front of 
and a little below the slit, examine the spectrum, and 
determine the wave-lengths of the lines. From Tables 31 
and 32 find what metals were present in the mixture. 

The spectra of the less volatile metals may be studied 
by allowing the spark from a Ruhmkorffs coil to play 
before the slit, the terminals being of the metal to be 
studied; the spectra of gases by viewing the discharge 
through vacuum tubes in which the gases are contained. 

THE DIFFRACTION SPECTRUM. 

If a ray of monochromatic light enter a darkened 
chamber at two parallel slits, S\ S' (Fig. 60), these slits 




Fig. 60. 



may be considered two sources of light. If any point, p, 
on a screes is farther from one of the openings than from 
the other by a distance S sin 0= I A, where S is the dis- 



SPECTRUM ANALYSIS. 115 

tance apart of the slits, and the angle between S' p, and 
the perpendicular from S upon the screen, and A. is the 
wave-length of the light employed, it is evident that 
the two rays will interfere at p, and darkness will result 
at that point. If, however, at a pointy', the difference in 
distance is A, the waves will re-enforce each other, pro- 
ducing a bright line. Thus a series of dark and bright 
lines will appear upon the screen. If, instead of mono- 
chromatic light, white light be employed, the points of 
re-enforcement will be at different distances from for 
light of different wave-lengths. The violet will be the 
least refracted and the red most. It follows that each 
of the bright bands obtained with monochromatic light 
will, in the case of white light, be spread out into a spec- 
trum, the order of the colors in which will evidently be 
the reverse of the order in the refraction spectrum. As 
we go outward from the centre (9, the bands increase 
in width until they begin to overlap, so that beyond the 
third or fourth order they are too confused for use in 
spectrum analysis. The spectrum obtained from a single 
pair of slits is very faint. By employing a number of 
slits equidistant from each other, the brightness may be 
much increased. A glass plate ruled with a large num- 
ber of equidistant parallel lines, with transparent spaces 
between them, is called a grating, and the spectrum 
obtained with it is called a diffraction spectrum. From 
the fact that the deviation is proportional to the wave- 
length, it is often called the normal spectrum. 

Exercise 63. — To determine S, the constant of a grating. 

By the constant of a grating is meant the average dis- 
tance between the lines. 

Adjust the spectrometer for parallel rays as in 



116 



I'll TSIt '. 1 L M EAS UREMENT. 



9 L 




Fig. 61 



(1) 8 = 



Exercise 00, and place the grat- 
ing upright upon the table (see 
Fig. 61) as nearly as possible 
at right angles to the ray of 
sodium light, by fixing the tele- 
scope upon the first bright band 
to the right, and turning the 
table slightly till the position of 
minimum deviation is obtained. 
Next set the telescope upon the 
central bright band, and then 
on the first, second, third, and 
fourth successively, and mea- 
sure the angular distances X , 
2 , 3 , # 4 , from the central band. 
Then — 



2X 

sin 6 



3X 



sin 



nX 

sin n 



sin X 

For A Xa , see Table 33. 

The grating is often ruled upon a reflecting surface of 
metal. In this case the reflected ray is observed. 



Exercise 64. — To determine the -wave-length of a line of the 
spectrum. 

From (1) it follows that — 

X = 8 sin n / n. 

Determine the wave-lengths of the principal lines of 
the spectrum of lithium in this way. 



POLARIZATION. 117 

POLARIZATION. 

Polarized light differs from ordinary light in that the 
vibrations in the polarized ray are confined to a single 
plane. Certain substances, of which Iceland Spar is the 
best example, have the property of quenching all the 
vibrations except those taking place in two planes which 
are at right angles to each other. The velocities of the 
light in these two planes are different, hence one ray is 
refracted more than the other. This fact is made use of 




Fig, 62. 

in Nichol's prism, which is a device for obtaining a ray 
of polarized light. It consists of a crystal of Iceland 
spar cut diagonally in two at CB as shown in Fig. 62, and 
the parts cemented together again with Canada balsam. 
When a ray of light, ab, enters the prism at b, it is divided 
into two rays, be, vibrating at right angles to the plane of 
the paper, and be, vibrating in the plane of the paper. 
The ordinary ray, be, is most refracted, and meets the 
balsam surface at e, at an angle a little greater than the 
critical angle for the two substances, and is hence totally 
reflected to d. The extraordinary ray, be, passes directly 
through the prism and out at d r . If a second Xichol be 
placed in line with the first, having the plane ABCD par- 
allel to the same plane of the first prism, a ray of light 



118 PHYSICAL MEASUREMENT. 

which lias been plane polarized passing through the first 
prism will also pass through the second. If the second 
prism be turned at right angles to the first, the ray will 
enter it in the plane of the ordinary ray, and be reflected 
out so that none of the light will pass through. For any 
other angle there will be a component in each plane, hence 
a portion of the light will pass through. 

ROTATION OF THE PLANE OF POLARIZATION. 

"When the two prisms of a polariscope are crossed so 
that no light passes through, it is found that, on placing 
certain substances between the prisms, a portion of the 
light will pass. Such substances are said to rotate the 
plane of polarization. The amount of rotation may be 
measured by determining the angle through which the 
analyzer, #, of the polariscope must be turned after insert- 
ing the substance to again extinguish the light. The sub- 
stance to be investigated is held, if a solid, in a clamp, if 
a liquid, in a tube which is closed at the ends with parallel 
glass plates. As the rotation is different for light of dif- 
ferent wave-lengths, being for most substances greater for 
the violet than for the red, it is necessary to use monochro- 
matic light. Substances which rotate the plane of polar- 
ization to the right are called dextrorotary, those which 
rotate to the left, leevorotary. 

Exercise 65. — To determine the percentage of sugar in a 
solution. 

Cross the prisms, direct the instrument at a sodium 
light, and insert tin; tube empty to see if there is any rota- 
tion due to tl ii3 glass plates. Read the vernier for the 
position of darkness, remove the tube, and fill it with the 
solution, being careful to leave no air-bubbles in the tube. 



POLABIZATIOX. 119 

Insert the tube again, and take a second reading for the 
position of darkness. The difference, a, of the two read- 
ings is the rotation for a column of the solution of length 
I cm. The specific rotary power of the solution for sodium 
light is — 



a 
Id 



where I is the length in decimetres, and d the density of 
the active substance in the solution. If we multiply this 
value by the molecular weight, ???, of the substance, then 
m [a] represents the molecular rotation. For convenience 
this value is usuallv divided by 100, so that — 

r -, ma 

\m\ = . 

L J 100 Id 

If there are p grams of the active substance dissolved 
in T'c.c. of the solvent, and the latter is supposed to be 
wholly inactive, — 

\a\ = , and \m\ = • . 

L J Ip ' L J 100 Ip 

To determine T^ weigh the tube empty, and again when 
filled with water. 



PART II (A). 

DISCUSSION OF RESULTS. 



ERRORS. 



Errors of observation may be divided into two general 
classes : — 

1. Constant errors, which are due to (ci) faults in the 
method ; (5) faults in the apparatus ; (c) personal error 
(that is, a tendency on the part of the observer to commit 
certain particular faults of observation), and — 

2. Accidental errors. The first class can evidently 
only be avoided by varying the apparatus, the methods, 
and the observers. The second class, in which the errors 
are the result of accident, may be treated by the theory of 
probabilities. This shows that, if the first class of errors 
has been eliminated, any observed value is as likely to be 
above the true one as below it ; hence, if a large number of 
observations have been taken of the same quantity, it is 
probable that the variations have been just as many and as 
great above as below the correct value. In this case, then, 
when the same quantity has been observed a number of 
times, the arithmetical mean of the observations gives the 
most probably correct result. If, upon inspection, the dif- 
ferences between the single observations and this average 
are found to be large, it is evident that the probable 

121 



1'2'2 PHYSICAL MEASUREMENT, 

accuracy of the work is loss than it would have been 
if the differences had been small. It is shown in works 
on least squares that if n be the number of observations, 
(In <l 2 < d Zi etc., be the differences between the single ob- 
servations and the average, and 2d 2 represent the sum 
of the squares of these differences (residuals), the prob- 
able error of the average, — 



= ±»v^ 



%d" 



n (n — 1) 
and the probable error of a single observation ■ 



V n — 1 



Example : — 

Suppose the time of swing of a torsion pendulum has 
been determined ten times, as follows: — 



TIME OF SWING. 


d 


d 2 


22.32 sec. 


+ 0.001 


0.000001 


22.31 


+ 0.011 


0.000121 


22.35 


- 0.029 


0.000841 


22.30 


+ 0.021 


0.000441 


22.34 


-0.019 


0.000361 


22.32 


+ 0.001 


0.000001 


22.32 


+ 0.001 


0.000001 


22.33 


- 0.009 


0.000081 


22.29 


+ 0.031 


0.000961 


22.33 


- 0.009 


0.000081 


Average 22.321 




2d 2 = 0.002890 



The probable error of the average 



■•' V 910 

The prol)al>le error of a single observation — ■ 



= ±fv /W« ± 0.0U9. 






ERRORS. 123 

The time of swing of the pendulum is then 22.321 i 

0.0038 sec. That is. the result is probably uncertain to 
the extent of Jt 0.0038 see. The calculation of the prob- 
able error has but little value unless the number of obser- 
vations is considerable (ten or more). 

The probable error furnishes information in regard to 
the accuracy of the work as far as it depends upon the 
accidental errors of observation, but none at all in regard 
to the constant errors. If these last have been eliminated, 
the accuracy of the result is inversely proportional to the 
square of the probable error. 



APPROXIMATE FORMULAE FOR CALCULATIONS WITH 
SMALL QUANTITIES. 

Often in the discussions of results the mathematical 
expressions are found to contain quantities very small in 
comparison with others under consideration. If these 
small quantities be squared, the square is as much smaller 
than the quantity as this is smaller than unity. Consider- 
ing this, it is evident that in many cases it will be allow- 
able to neglect the squares and higher powers, as well as 
the products of such quantities. Thus, while 0.001 of 
a unit may be important. 0.000001 may be disregarded. 
When the expression can be brought into the form 1 plus 
minus the small quantity, the following formulae can 
be used to advantage. When the sign ^ or ^f is placed 
before a quantity, it must be used with either the upper 
or lower sign throughout the formula. 

Let 8. e. 4. ?; represent fractions of unity so small 
that their squares, higher powers, and products may be 
neglected.* 

* These formula? are taken from. Kohlrausch's Praktische Physik. 



1-4 PHYSICAL MEASUREMENT. 

(1) (1 + 8)'" = 1 + W 8; (1 - 8)'" = 1 _ m & 

(1+8)" =1 + 28; (1-S) 2 =1-28. 

(3) a F+l = 1 + £8; Vl^S = 1 - i 8. 

» iti =1 - Si rh -* + < 

1 1 

(6) , = 1 - i- 3 ; =l + i& 

(7) (i±«)Ci±«)(l±0...=l±*±c±C... 

W (1 ±c) (1 ± ,) . . ■ ^ ^ 4 T T * 

If two quantities, P x and _P 2 , differ only slightly from 
each other, so that P 2 = P x -f 8, their arithmetical mean 
may be used instead of their geometrical : — 

(9) Vap 2 ^ *J 2 ; 

also — 

(10) sin (X + 8) = sin X + 8 • cos X sin 8 = 8 ; 

(11) cos (X -f- 8) = cos X — 8 • sin X cos 8 = 1; 

(12) tan (X + 8) = tan X -\ — tan 8 = 8: 

the unit angle being 57°. 3, for which the arc equals the 

radius. 

DETERMINATION OP CONSTANTS BY THE METHOD OP 
LEAST SQUARES. 

If the result sought is not measured directly, but is 
connected with the measured quantities by an equation, 
the method of least squares furnishes the most accurate 
method of determining the values of the constants in the 
equation. This method is based on the assumption that, 
in a series of observations, small errors are more likely to 



ERRORS. 125 

occur than large ones ; and the more accurate the obser- 
vations, the less will be the proportion of large errors to 
small. It is shown, on this assumption, by the theory of 
probabilities, that the accuracy is inversely proportional 
to the sum of the squares of the errors. The problem to 
be solved by least squares is in general to determine the 
equation of a curve, which, if the results were platted 
graphically, would run through them so that the sum of 
the squares of the distances of the single observed points 
from the curve would be a minimum. In all the cases 
time will be saved in the somewhat laborious calculations, 
and the accuracy of the work will not be lessened, if, when 
it is convenient to do so, instead of taking a large number 
of observations of different values of y, corresponding to 
different values of #, a large number of observations be 
taken on each of the ys corresponding to perhaps three 
or four values of x, chosen as far from each other as is 
convenient. Then the arithmetical mean of each of the 
^/'s may be found, and these mean values used in the cal- 
culations (see following cases). 

Case I. Suppose we observe several values of a quan- 
tity, which is connected with another quantity, #, by the 

equation, — 

y = ax-, 

and it is required from these observations to determine 
the value of the constant a. The method of least squares 
states that the most probable value of (a) is that which 
makes the sum of the squares of the errors, or — 

(1) fa - ax x y + (y 2 - ax 2 ) 2 + (*/ 3 - ax z ) 2 + etc. ; 
or, 2 (y — ax) 2 = a minimum. 

From the calculus this is a minimum when the so- 
called "normal equation" — 

(2) 2 (y — ax) x = 



126 PHYSICAL MEASUBEMENT. 

is satisfied. This gives for (<*) the probable value — 
(3) a = Say/Sa 9 . 

If the value of .r is constant, we have a special case in which we 
ran introduce instead of ax a constant c. This gives, instead of Equa- 
tion 2, — 

2" (yx — ex) = 0; or 

3 7/.r = ncas, 

where n is the number of observed values of ?/, then — 

(4) c-& 

n 

(the arithmetical mean) 

Example: — 

To find the constant (a) of an electro-dynamometer. 
The current C is connected with the angle of deflection 
by the equation, — 

C 2 = a - sin \p. 

Suppose the current corresponding to deflections of 
20°, 30°, 40°, 50°, has been found a number of times, 
and that the square of the means of these readings is 
given in the second column of the following scheme of 
calculations : — 



$ C 2 SIN*/' C 2 -SIN^ 


SIN 2 */' 


20° 0.099 0.342 0.0339 


0.1170 


30° 0.152 0.500 0.0760 


0.2500 


40 0.194 0.643 0.1247 


0.4134 


50 D 0.230 0.766 0.1762 


0.5868 


SO 2 - sin * = 0.4108 ^sin 2 «A = 


= 1.3672 


From Equation 3, 

^C 2 -sin^ 0.4108 A onA 

a = — = = U.oUU. 

2sin 2 * 1.3072 




The equation thus becomes — 




C 2 = 0.300 • sin ^. 





Calculating C 2 from this formula, and finding the 
deviations, cZ, of the observed values, we have: — 







ERRORS. 






BSERVED. 


CALCULATED. 








C 2 


a. sin ^ 


d 




d 2 


0.099 


0.1026 


+ 0.0036 




0.00001296 


0.152 


0.1500 


- 0.0020 




0.00000400 


0.194 


0.1929 


- 0.0011 




0.00000121 


0.230 


0.2298 


- 0.0002 


2d* 


0.00000004 
= 0.00001821 



127 



If any other value be taken for (a) it will be found 
that 2c? 2 is increased. 

Case II. If the quantities are related by the equation 

(5) y = c + ax, 

and a number of values of y have been observed, which 
correspond to different values of .r, by least squares the 
most probable values of c and a are those which make — 

2 \y — (c -\- ax)] 2 = a minimum. 
From the calculus this equation will be true if the two 
normal equations are satisfied, — 

(6) ai>- (* + «*)]■= o ; 

2 \_xy — (ex + ax' 2 )'] = ; 

or, more conveniently written, 

2^/ — nc — a^x = ; 
and %xy — c%x — a%x L = ; 

Where n is the number of observed values of y. From 
these — : 

__ %x-%xy — 2,z/,2x- 2 . 
&x) 2 -n%x 2 ; 

(2x) 2 - ?^^ 2 " 
To diminish the difficulties of calculation with large 
quantities, it is allowable to subtract from all the values 
of y or x a constant quantity; and these remainders will 
represent the excess of the y's or x's above the quantity 
subtracted. This subtracted quantity must be again intro- 



128 PHYSICAL MEASUREMENT. 

duced in the final calculations. The amount subtracted 
should be so chosen that it equals, or nearly equals, the 
smallest value of y or #, or one of the values near 
the middle of the series. In this last case some of the 
remainders will he positive and some negative. This is 
graphically equivalent to a parallel displacement of the 
co-ordinate axes. 

EXAMPLE, Case II. (From Kohlrausch's Praktische 
Physik.} 

To find the coefficient of expansion of a metal bar. 
By comparison with a standard, the length was observed 
to be as follows : — 



TEMPERATURE 


LENGTH. 


TEMPERATURE. 


LENGTH. 


20° 


1000.22 mm. 


50° 


1000.90 mm 


40° 


1000.65 


60° 


1001.05 



The formula for the length, ?, at any temperature, £, is 
I = l [1 + $ (t - t )l 
where l is the length at the original temperature, £ , and 
/3 is the coefficient of expansion. If t is taken as 0°, the 
formula becomes — 

I = h (1 + Pt). 

Letting l (3 = a, — 

I = l -\- at, 

which is an equation of the same form as (5). 

To simplify calculations, 1000 is subtracted from all 
the lengths. The remainder R = y — 1000. 



t 




It 


t* 


tB 


20° 




0.22 


400 


4.4 


40° 




o.r>5 


1600 


26.0 


50° 




0.90 


2500 


45.0 


00° 




1.05 


3600 


63.0 


2t = 170° 


SB 


= 2.82 


Zt' 2 = 8100 


2tR = 138.4 



ERRORS. 129 



From Equation (7), 



t 


l OBSERVED. 


l CALCULATED. 


20° 


1000.22 mm. 


1000.228 mm, 


40° 


1000.65 


1000.652 


50° 


1000.90 


1000.864 


60° 


1001.05 


1001.076 



7 170.138.4-2.82.8100 AiM 

l ° = 170- - 4 ■ 8100 =-°' 196 ; 

a = 170-2.82-4-138,4 = aom 

1702 _ 4 • 8100 T 

The length of the bar at 0°, adding the 1000 mm. 
which was subtracted, is — 

l = 1000 - 0.196 = 999.804 mm., 
and the length at any temperature, % is — 
I = 999.804 + 0.0212 t ; 

calculating the lengths for 20°, 40°, 50°, and 60°,— 

d d 2 

+ 0.008 mm. 0.000064 

+ 0.002 0.000004 

- 0.036 0.001296 

+ 0.026 0.000676 

2d 2 = 0.002040 

Any variation in the values of l and a will be found to 
increase 2^ 2 . The coefficient of expansion, — 

£ = — = 0.0000212. 

If it is known for certain that the equation between the two vari- 
ables has the form y = c + ax, that is, that the curve represented is 
a straight line, it is possible to determine the values of the constants 
with considerable accuracy without the use of least squares. Two 
values of x, x x , and x 2 , are taken as far from each other as possible, 
and a number of observations made on the corresponding values of y, 
2/j , and y. 2 . Let the mean of the values found for y ± be ?//, and for y 2 
be 2/ 2 '. Then, — 

c = V\v4 — x<iV\ . a= vi — y* 

Xi — Xo \ — ° 

Case III. The equation here is, — 

(8) y = c -f- ax -\- bx 2 , 



130 PHYSICAL MEASUREMENT. 

and the values of the constants, e, a, and b are to be so 
determined that — 

^ \_U — ( c + ax + foe 2 )] 2 = a minimum. 
From the calculus we get the three normal equations, — 
%y — nc — a%x — b^x' 1 = ; 
(9) %xy - c%x - a^x 2 - b$x* = ; 

%x 2 y - c%x 2 - a%x z - b%x± = 0. 

The values of the constants a, 5, and <?, can be derived 
from these three equations. 

Example, Case III. (From Uppenborn's Kalender 
fur Electrotechniker, 1891.) 

The intensity of light given by a certain gas-burner is 
to be determined as a function of the height of the flame. 
The following values were found, each value of the inten- 
sity being the mean of several observations : — 



INTENSITY. 


HEIGHT OF FLAME. 


INTENSITY. 


HEIGHT OF FLAME. 


V 


X 


y 


X 


0.838 


51 


1.143 


71 


0.965 


61 


1.369 


81 



Inspection shows that y increases more than propor- 
tionally to x; hence, it may perhaps be represented by 
Equation (8). To simplify the calculation, instead of x, 
we will introduce x l + 61. 



x x 


y 


x i 


x x s 


x£ 


*iV 


x?y 


-10 


0.838 


100 


-1000 


10000 


- 8.38 


83.8 





0.005 

















+ 10 


1.143 


100 


+ 1000 


10000 


+ 11.43 


114.3 


+ 20 


1.300 


400 


+ 8000 


160000 


+ 27.38 


547.6 



20 4.315 600 8000 180000 30.43 745.7 

Introducing these values into the Equations (9), — 

4.315 - \a x - 20 b x - 600 e x = 0. 

30.4:; _ 20 a, - 600 6, - 8000 c, = 0. 

745.7 _ coo a , _ 8000 6, - 180000 c x =-. 0. 



ERRORS. 131 

From these, 

«! = 0.9677, b y = 0.01546, e x = 0.000225, 
y = 0.9677 + 0.01546 x, + 0.000225 xf. 

Or, again introducing x = x x -f 61, — 

y = 0.9677 + 0.01546 (x - 61) - 0.000225 (x - 61) 2 ; 
y = 0.861865 - 0.01199 x + 0.000225 cc 2 . 

Calculating y from this formula, — 



OBSERVED. 


CALCULATED. 


a 


a a 


0.838 


0.8356 


- 0.0024 


0.00000576 


0.9G5 


0.9677 


+ 0.0027 


0.00000729 


1.143 


1.1448 


+ 0.0018 


0.00000324 


1,369 


1,3669 


- 0.0021 


0.00000441 




0.00002070 



The probable error in results calculated by least 
squares is expressed as follows : — 

The probable error of the single observed values of 

y is — 

3 V n — m 

of the calculated values is — 



2 / 2d 2 

: 3 V n ( n _ 



(n — m) 

where n is the number of observations, and m is the 
number of constants determined. 

For further directions for the use of least squares, cal- 
culations with approximate values of the constants, Gauss' 
Method, etc., see Kohlrausch's Praktische Physik. A 
still more extensive treatment of the subject may be found 
in the treatises of Comstock and Merriman, 



PART II (B). 

MANIPULATIONS, 



MANIPULATIONS WITH GLASS. 



To cut glass. — To cut window or plate glass in any 
rectilinear shape, lay a large sheet of paper on a flat table, 
and lay off on the paper the size and form desired. Place 
the glass over the paper, and hold a ruler firmly upon it, 
a little to the left of the line to be cut. If a diamond is 
used, it must be held at exactly the same angle through- 
out the cut; and the pressure must not be great enough 
to cause the glass to crackle and produce a wide mark, 
the finest scratch possible being sufficient. The diamond 
must never be drawn twice over the same line. If a com- 
mon steel-wheel cutter is used, more pressure must be 
exerted than with the diamond, and the scratch must be 
deep enough to be plainly visible. After making the cut, 
grasp the pane firmly near one edge, with the thumb and 
Hi id finger at both sides of the scratch and as near to it 
as possible, and press as if to open the crack. It should 
break on the line. If the plate is large, place it so that 
the scratch lies exactly over the edge of the table, grasp 
the overhanging edge with both hands, lift the pane 
slightly from the table, and bring it down with a quick 
motion, which will snap it off squarely, even if the scratch 

132 



MAN IP ULA TIONS. 1 3 3 

is not a good one at all points. Should any irregular pro- 
jections remain, they may be removed by means of the 
notches usually found at the sides of the glass-cutter ; or, 
much better, with a pair of good end-cutting nippers. 

The nippers serve well also for cutting small circles 
or irregular figures, after they have first been blocked out 
with the cutter. 

To cut off a glass tube, make a clean cut across the 
tube with the corner of a sharp file, grasp the tube firmly 
in both hands, with the ends of the thumbs opposite the 
cut, and bend towards the thumbs. If the tube is a large 
one, make the cut deep, heat the end of a small glass rod 
red-hot in the flame, and press it firmly at one end of the 
cut. The glass will usually break off squarely. If the 
tube be very large, it may be necessary to draw the hot rod 
along the course it is desired that the crack should take, 
keeping just ahead of the crack. An iron rod may be used 
instead of a glass rod. The end of a large tube may be 
smoothed with a fine file or whetstone. The end of a 
small tube should be held in the flame till it begins to 
soften, which will be shown by its taking a yellow color, 
due to the sodium in the glass. 

To shape glass by the flame. — For most of the ordi- 
nary manipulations, the common Bunsen flame will answer. 
For large tubing, a large-sized Bunsen or blast lamp saves 
much time. For heating a small spot, as in making "J" 
joints, the pointed flame of the blast-lamp is almost a 
necessity. 

Glass should never be brought suddenly into a flame. 
If it is introduced and withdrawn repeatedly, and kept 
constantly in motion, it will seldom crack. If it is to be 
tough after manipulation, it should never be cooled 
directly from the hot flame, but held for a few^ seconds in 



134 PHYSICAL MEASUREMENT. 

the smoky flame made by excluding the air from the 
Bunsen burner, till it is cool enough so that soot begins 
to collect upon it. Glass should never be bent or blown 
while in the flame. Keep it turning between the fingers, 
and it may be heated to any desired softness. It may 
then be taken from the flame, and manipulated at pleas- 
ure. To close the end of a small tube, hold it in the 
flame until it closes itself. To close a larger tube, heat 
the end, and draw the edges together by means of a cool 
glass rod, to which the hot glass will adhere ; pull the rod 
away suddenly, and a fine tip will remain, which may be 
removed by heating, and touching to the cold rod. If a 
small thick spot still remains at the end of the tube, heat 
it red-hot, remove from the flame, and blow into the open 
end. 

To join the ends of two glass tubes. — To join the ends 
of two glass tubes of different sizes, close the end of one 
of the tubes, heat it red-hot, remove from the flame, and 
blow a small uniform bulb, twirling the tube to prevent 
the bulb from settling out of shape by its own weight. 
Heat the little bulb again and blow quickly till it is so 



Fig. 63. 

thin that it bursts, brush away the thin portions, leaving 
the end as shown in Fig. 63. 

The other tube is to be treated in the same way, after 
which both ends are heated at once, pressed against each 
other until they unite, and then drawn apart a little to 
reduce the size of the joint to the size of the larger tube. 
If the tube contracts so as to make the bore too small, 



MAN IP ULA TIONS. 135 

close the large tube with a cork, and enlarge the joint by- 
heating and blowing. 

To make a T joint, close one end of one tube, heat a 
small spot on its side, and blow a bulb there so as to give 
an opening with edges suitable for joining. The rest of 
the process is similar to that just described. Success in 
joining glass is often conditioned upon the glass being all 
of one sort. Even the trained glass-blower often fails 
when trying to work with glass of two very different 
kinds. 

To bend a glass tube, close one end, heat the place to 
be bent, remove from the flame, bend part way, heat and 
blow till of uniform size, heat a little to one side of the 
first bend and bend again, alternating bending and blow- 
ing till the desired result is obtained. A good-looking U 
tube of any considerable size is by no means easy to make. 
For small tubes, where a slight contraction in size is no 
objection, heat quite a long portion of the tube, and bend 
at once without blowing. 

To bore glass. — To bore a small hole in a glass plate, 
wet the spot with a mixture of turpentine and camphor, 
and bore with the end of a file. Large holes may be 
bored by means of a brass tube and emery powder wet 
with water. The tube should be held in place by boring 
a hole the size of the tube in a board, and cementing the 
board to the glass plate. 

Silvering glass. — The following method is recom- 
mended for producing a firm, adherent, and brilliant coat- 
ing of silver on glass. The two solutions necessary for 
this process are made as follows : — 

The reducing solution. — Dissolve 100 grams rock 
candy or pure loaf sugar in 800 c.c. distilled water, then 
add 5 c.c. nitric acid and 200 c.c. alcohol. This reducing 



136 PHYSICAL MEASUREMENT. 

solution will keep a long time, and, indeed, improves with 
age. It should be made at least a day or two before 
using. 

The silver solution. — Dissolve 5 grams caustic potash, 
pure by alcohol, in 100 c.c. distilled water. Dissolve also 
in a separate beaker 10 grams of nitrate of silver in 100 
c.c. distilled water, and set apart about one-sixth of this 
solution as a reserve. To the remaining five-sixths of the 
silver solution add ammonia drop by drop, until the dark- 
brown precipitate of oxide of silver which is formed at 
first is just dissolved, and the solution becomes clear. 
Into this mixture now pour the potash solution mentioned 
above, and the liquid will become nearly black. Then 
add ammonia once more drop by drop, until the precipi- 
tate is dissolved, and the solution again becomes clear. 
Silver solution must now be added from the reserve until 
a permanent, flocculent, dark-brown precipitate is formed. 
The solution is then filtered, and the light-brown or 
straw-colored filtrate is ready for use. 

Much of the success in silvering depends upon the 
thoroughness with which the glass surface to be silvered 
is cleaned. This is best effected by pouring nitric acid 
upon the glass, and rubbing the surface with a tuft of 
absorbent cotton or filter paper tied to a glass rod. The 
nitric acid is then washed off, and a solution of caustic 
potash poured on, and the surface again rubbed with 
another tuft of cotton. The glass must then be washed 
thoroughly with distilled water, after which care must be 
taken not to touch the surface with the fingers. 

The glass cleaned in this way is then placed in a glass 
or porcelain vessel, with the surface to be silvered upward. 
The light^brown or straw-colored filtrate mentioned above 
is then poured over the glass. If the latter is not entirely 



MAX IP UL A TI0N8 . 137 

rered by the solution, distilled water must be added 
until the glass is completely submerged. Fifty o.e. of the 
reducing solution is then added: and the vessel containing 
the glass to be silvered must be gently rocked back and 
forth during the time in which the silver is depositing. 
which will be about five minutes. At first the solution 
will turn intensely black, but will gradually otow lighter 
until toward the end of the operation, when it will be 
light gray. A muddy gray precipitate will also form, and 
it is to keep this' from settling upon the glass that the 
vessel must be rocked. The light-gray color of the solu- 
tion indicates the end of the silvering process. The glass 
must then be taken out. washed, and placed upon its edge 
to dry. When perfectly dry, it can be rubbed quite hard 
with a piece of chamois skin, and later polished with a 
piece of chamois skin which contains a minute quantity 
of rouge. 

The quantities here given are sufficient for a surface 
15 or 20 cm. square. 

If these instructions have been followed carefully, 
there should result a hard, firm coat of silver, which will 
not rub off when polished, and through which the sun 
appears of a bright-blue color. 

The following solutions * will perhaps sometimes be 
found more convenient than the above, as they may be 
kept ready for use for a long time, if kept in the dark; 
our own experience has been, however, that the results 
are not as uniformly good as those given by the first 
method. 

(a) Five grams of silver nitrate are dissolved in about 
forty c.c. of distilled water, and ammonia is added until 
the precipitate at first thrown down almost entirely dis- 
* From Kohlraiiseli. Praktische Physik. 



138 PHYSICAL MEASUREMENT. 

appears. The solution is then filtered, and diluted to 
500 c.c. 

(/>) One gram of silver nitrate is dissolved in a little 
hot distilled water, and this is poured into one-half litre 
of rapidly boiling distilled water. In this dissolve 0.83 
gram of Rochelle salt, and boil about twenty minutes 
till the precipitate becomes gray, and filter while hot. 

The glass to be silvered, after being cleaned as in the 
first method, is placed in a glass dish, and covered to the 
depth of a few mm. with equal volumes of the solutions 
(#) and (6). The silvering will be completed in about 
one hour. 

QUARTZ FIBRES. 

By far the best suspensions for galvanometers, magnet- 
ometers, electrometers, and like instruments, are quartz 
fibres, the remarkable qualities of which were first discov- 
ered a few years ago by Professor C. V. Boys. They are 
far superior to silk, inasmuch as they have such perfect 
elasticity that the shifting of the zero, so often seen in silk 
suspension instruments, is entirely done away with. Their 
strength is remarkable, being greater than that of any 
other known substance. 

Drawing the Fibres. — The following very simple 
method of drawing these fibres is taken, with slight 
alterations, from Professor E. L. Nichols's published 
lectures on The Galvanometer. The apparatus consists 
of an oxyhydrogen blowpipe (the burner from a calcium 
lantern may be used), two pairs of crucible tongs, and 
some bits of white quartz. The quartz should be 
crushed into granules about three or four millimetres in 
diameter. These are placed in a cavity hollowed out in 
a block of charcoal, and, by means of the flame, melted 
together into a solid mass This may now be placed in 



MANIPULATIONS. 139 

the flame and manipulated, without the danger of the dis- 
integration which would have taken place if the crystals 
had been used in their natural state. The quartz mass 
is placed in the hottest part of the flame, and by means 
of the tongs manipulated first into a short rod and then 
drawn out into a fibre. The size of this fibre will depend 
on the quickness of the jerk with which the drawing is 
done. These coarse fibres may be used for many pur- 
poses ; and they may be reduced to any desired fineness 
by the simple process of holding the ends in the flame 
until they soften, when the rush of the heated gas from 
the burner carries them up with it, spinning them out 
to a fineness suitable for the most delicate instruments. 
These fibres, which are too small to be readily seen, may 
often be caught as they float up by the larger filaments 
which they drag behind them. Another method of cap- 
turing them is by means of a large piece of canton 
flannel, placed at a safe distance above the flame, the 
fibres catching upon the rough surface. 

The best method of preserving the fibres is by wind- 
ing them upon a frame made of two glass tubes be- 
tween two pieces of lath. The fibres are kept in place 
by rather thick shellac varnish put on the tubes just 
before the fibres are wound on. The length of the 
frame will depend upon the length of the fibres desired 
for use. 

Mounting the Fibres. — A bit of universal wax (see 
p. 142) is pinched* on to each end of the fibre before 
it is taken from the frame ; by means of these it may be 
handled, and kept in position. A little melted shellac is 
placed on the points to which the fibre is to be fastened. 
The fibre is held so as to rest against the shellac, and 
a touch from a hot iron causes this to melt and envelop 



140 PHYSICAL MEASUREMENT. 

the fibre. Owing to the molecular changes going on 
in the shellac, there may be a slight drift to -the zero 
of the instrument for the first clay or so after the fibre 
is hung. 

Quartz fibres are most excellent insulators. But if it 
is desired to make them conductors for any reason, for 
example, to make connection for an electrometer needle, 
they may be cleaned and silvered according to direc- 
tions for silvering glass (p. 135). When dry they may 
have their ends copper-plated by dipping them in a 
solution of copper sulphate, while the positive wire from 
a battery is in the solution and the negative is held in 
contact with the fibre. The coppered ends can now be 
soldered * to bits of copper foil or small copper wires. 

SOLDERING. 

The secret of success in soldering is to have both 
tool and work clean and hot. 

If the metal surfaces are new and bright, a little 
pulverized rosin will cause them to unite readily. If 
they are oxidized or oily, clean them just before apply- 
ing the solder with a solution of zinc chloride, made 
by dissolving bits of zinc in hydrochloric acid till the 
acid will dissolve no more. The surfaces should fit as 
closely as possible upon each other, and be held firmly 
together till the solder sets. Small objects, as two 
wires, may be heated in a Bunsen flame till the solder, 
which may be had in the form of wire for such purposes, 
melts when touched to it. If a soldering-iron is to be 
used, see that it is clean and well "tinned" with solder, so 
that it will readily pick up the solder. It will impart its 

* For complete directions for soldering quartz fibres, see C. V. Boys's 
" Attachment of Quartz Fibres," Phil. Mag., May, 1894, p. 4(53. 






MANIPULATIONS. 141 

heat much more readily also when thus coated than when 
covered with rust. If the iron has become rough from 
overheating, time will be saved by "trueing it up" with 
the file. Copper, brass, iron, in fact, all the metals com- 
monly employed in the laboratory, except aluminum, may 
be soldered to each other without difficulty. With lead 
and with zinc in thin sheets care must be taken not to melt 
the metal. In soldering large pieces of metal which cannot 
be conveniently heated otherwise, the hot iron should be 
held upon the joint to heat it, and the iron again heated 
before the solder is applied. Until iron and work are hot 
enough so that the solder flows freely, any attempt to 
make a joint is a waste of time. If the work is held in 
an iron vise, it should be kept away from the iron by 
strips of wood, otherwise the large mass of metal will 
conduct away the heat. 

If a long joint is to be made, the pieces may be held 
together at one or two points by drops of solder, while the 
remainder is soldered. 

If the joint when made is a strong one throughout, but 
not smooth, go over it slowly with a hot iron. 

CLEANING MERCURY. 

Commercial mercury is comparatively pure chemically. 
It may be freed from mechanical impurities by filtering 
through a piece of paper which has several pin-holes 
pricked through it at the point, or by holding a large 
card firmly against the edge of a small evaporating- 
dish, and allowing the mercury to flow out under the 
card. 

Mercury combines with zinc and lead so readily that 
it is likely to contain these metals as impurities after 
being used in the laboratory. These may be removed by 



142 PHYSICAL MEASUREMENT. 

washing with dilute nitric acid, or better, with a solution 
of 5 g, potassium bichromate and 5 c.c. sulphuric acid in 
one litre of water, and drying with filter paper. 

The only method of obtaining absolutely pure mer- 
cury is by distillation in a vacuum, for which purpose 
a specially constructed glass still is required. 

The free sulphur contained in the rubber tubes, used 
in Geissler pumps and other apparatus, frequently con- 
taminates the mercury in its passage through them. This 
may be prevented by coating the inside of the tube with 
the rubber solution described below. 

CEMENTS AND VARNISHES. 

1. For Glass. Two parts shellac dissolved in 1 part 
turpentine, and cast into sticks. 

2. Hard Wax. Equal parts beeswax and rosin melted 
together. 

3. Universal Wax (soft). Melt together 1 part Venice 
turpentine and 5 parts beeswax. Color with vermilion. 

4. Eubber Cement. Burgundy pitch 2 parts, beeswax 
1 part, gutta-percha 1 part. Melt the pitch and beeswax 
together, add the gutta-percha in thin strips, and boil till 
it is melted. 

5. Photographers' Paste. Dissolve 4 ounces of gela- 
tine in 16 ounces of water, add 1 ounce of glycerine and 5 
ounces of alcohol. 

This is excellent for mounting paper scales on wood, etc., 
as it does not curl the paper. It is well suited also for use 
in prisms, for carbon disulphide. Set the vessel containing it 
in hot water a few minutes before it is wanted for use. 

P>. Mat Black Varnish. A solution of sandarac in 
alcohol, mixed with fine lampblack ; dries without gloss, and 
becomes hard without being brittle. 

7. Rubber Solution. Dissolve pure gutta-percha in 



MAN IP ULA TIONS. 



143 



chloroform, shaking the bottle occasionally, till no more will 
dissolve ; decant, and keep well stoppered. 

8. Lacquer for Brass. Dissolve ^ pound pale shellac 
in 1 gallon alcohol ; decant or filter, and keep in a closed 
bottle, excluded from the light. Heat the brass to about 
100° C, and apply two coats in rapid succession, with a 
camelVhair brush. 

STANDARD CELL. 

A form of standard Daniell cell which is well adapted 
for general use in the laboratory with the electrometer can 
be easily made as follows : Nearly fill two small test-tubes 



o 


jOl 







: : : : 




i 


VI {ft 


i 

- 



v_y 



w 



Fig. 64. 



(see Fig. 64), the one with a saturated solution of pure 
zinc sulphate, and the other with pure saturated copper 
sulphate. The two test-tubes are now connected by 
means of a U shaped capillary tube. This is filled with 
water and the water blown out, thus leaving the inside 
of the tube wet. It is then inserted inside the test-tubes ; 
and the coating of moisture is sufficient to make electrical 
connection between them, while the diffusion of the two 



144 PHYSICAL MEASUREMENT. 

liquids is entirely prevented. The moisture cannot evapo- 
rate from the inside of the capillary tube, as, of course, the 
air is saturated. 

The positive electrode is formed by depositing electro- 
1\ tie copper from a solution of pure copper sulphate on 
a copper wire. The negative electrode is made of a strip 
of twice distilled zinc. The mouths of the test-tubes are 
stopped with corks through which the capillary tube and 
the electrodes run, the liquids are heated to expel the air, 
and the whole is sealed up, while hot, with marine glue. 

The advantages of this cell are, its ease of preparation ; 
its freedom from the likelihood of injury, as leaving it 



Fig. 65. 

short-circuited has no effect upon it, since its internal 
resistance is several million ohms ; and its small tempera- 
ture coefficient, which is practically negligible for the ordi- 
nary range of laboratory temperature. The E.M.F. is 
1.072 volts. If ordinary commercial salts and metals be 
used in its preparation, the E.M.F. of the cell may vary 
by two or three tenths of one per cent from the above 
amount. 

WATER BATTERIES. 

The water battery of Rowland consists of a series of 
zinc and copper pairs cemented to the lower side of a glass 
plate, each pair being so near to the next that when the 
tips of the metal strips are dipped in water a layer of water 
will be held between the strips by capillary action. 



MAN IP ULA T10JSTS. 



145 



The form about to be described can be as easily filled 
as Rowland's, and does not require filling again for several 
weeks. Cut the required number of zinc and copper strips 
12 cm. long and 3 mm. wide. Bend them by means of a 
form made by driving small nails in a block of wood in the 
shape shown in Fig. 65. Saw cuts 3 mm. deep and 28 mm. 
apart in strips of dry pine wood 8 mm. thick, 28 mm. wide, 
and 34 cm. long, ten cuts in one edge of each strip. The 
zinc and copper pairs are first soldered together and then 
slipped into the cuts, after which 10 strips are mounted, 
with the addition of a blank strip to hold the last row 




Fig. 66. 

in place in the grooved end pieces, fastened firmly with 
nails, and the whole inverted for a few minutes in a pan 
containing shellac varnish to the depth of 2 cm. to insure 
perfect insulation on the part of the wood. The battery 
is now completed by springing a 1 drachm homoeopathic 
vial over each pair (see Fig. 66). The metal strips are so 
spaced that they hold the bottles in place without any other 
support. The battery is filled without wetting the frame 
in the least, by setting it over a pan, and filling the pan 
with water to the depth of 6 cm. 



1-10 PHYSICAL MEASUREMENT. 

Ten such frames of 100 cells each, giving a potential 
of 1000 V. arc placed five deep in a portable rack of wood, 
which should have shallow pans of tin below each frame to 
prevent wetting the frame below. This precaution will 
not be needed if the bottles are allowed to dry off before 
placing them in the rack. 



PART IV. 

TABLES. 



Note. — Tables 4-13 and 29-31 are from Kohlrausch's Praktische 
Physik. Tables 44-50 are from the Smithsonian Tables. All 
values given in the Tables are for a temperature of 15° C, where 
another temperature is not specified. 



148 



PHYSICAL MEASUREMENT. 



1. Density of the More Important Elements. 



Element. 



Aluminum 
Antimony . . 
Arsenic . . . 
Barium . . . 
Bismuth . . 
Boron . . . 
Bromine . . 
Cadmium . . 
Calcium . . . 
Carbon . . . 

Diamond . 

Graphite . 

Gas Carbon 

Charcoal . 
Chlorine . . 
Chromium . . 
Cobalt . . . 
Copper . . . 
Fluorine . . 
Gold .... 
Hydrogen . . 
Iodine . . . 
Iron .... 

Wrought . 

Cast . . . 

Steel . . 
Lead .... 
Lithium . . . 
Magnesium 
Manganese . . 
Mercury . . . 
Nickel . . . 
Nitrogen . . 
Oxygen . . . 
Phosphorus 

Common . 

Bed . 

Metallic . 
Platinum . . 
Potassium . . 
Silicon . . . 
Silver . . . 
Sodium . . . 
Strontium . . 
Sulphur . . . 
Tin .... 
Zinc .... 



Symbol. 



Al. 

Sb. 

As. 

Ba. 

Bi. 

B. 

Br. 

Cd. 

Ca. 

C. 



CI. 

Cr. 

Co. 

Cu. 

F. 

Au. 

H. 

I. 

Fe. 



Pb. 

Li. 

Mg. 

Mn. 

Hg. 

Ni. 

N. 

O. 

P. 



Pt. 
K. 

Si. 

Ag. 

Na. 

Sr. 

S.- 

Sn. 

Zn. 



Atomic 
Weight. 



27.3 

122.0 

74.9 

136.8 

207.5 

11.0 

79.75 

111.6 

39.9 

11.97 



35.36 

52.4 

58.6 

63.3 

19.1 

196.2 
1. 

126.53 
55.9 



206.4 
7.01 
23.94 
54.8 

199.8 
58.6 
14.01 
15.96 
30.96 



196.7 
39.03 

28 
107.66 

23 

87.2 

31 .98 
118.8 

64.9 



Density. 



2.60 
6.71 
5.73 
3.75 
9.80 
2.57 
3.15 
8.60 
1.57 

3.52 
2.30 
1.89 
1.6 
1.47 
6.50 
8.6 
8.92 
0.00171 
19.32 
0.0000896 
4.95 

7.82 

7.65 

7.70 
11.37 

0.59 

1.74 

7.39 
13.55 

8.9 

0.001255 

0.00143 

1.83 

2.20 

2.34 
21.50 

0.87 

2.39 
10.53 

0.978 

2.54 

2.07 

7.29 

7.15 



TABLES. 



149 



2. Density of Solids (Not Elements). 



Substance. 



Density. 



Substance. 



Density. 



Asbestos 
Brass . 
Bronze 
Chalk 
Coal . 
Cork . 
Feldspar 
German SilYer 
Glass, common 

flint . 
Granite . . 
Gyp sum . . 
Ice . . . 
Iceland Spar 
Ivory . . . 
Marble . . 



2.1 

8.1-8.6 

S.7 

2.3-3.2 

1.4-1.8 

0.14-0.3 

2.55 

8.5 

2.5-2.7 

3.3-3.8 

2.5-2.9 

2.3 

0.917 

2.75 

1.9 

2.65-2.8 



Mica .... 
Paramne . . 
Porcelain . . 
Quartz . . . 
Rubber, soft . 

vulcanite 
Salt, rock . . 
Sand .... 
Sandstone . . 
Sugar . . . 
Tallow . . . 
\Yax .... 
Wood — 

Ebony . 

Oak . . 

Pine . . 



2.G5-2.93 

0.87-0.91 

1.71-2.38 

2.(35 

0.92-0.99 

0.97 

2.16 

2.8 

2.36 

1.58 

0.92 

0.96 

1.2 
0.7 
0.5 



3. Density of Liquids. 



Substance. 


Density. 


Substance. Density. 


Acids — Acetic . . . 


1.06 


Chloroform . . . 1.53 


Hvdrocbloric . 


1.16 


Carbon Distil phide 


1.29 


Xitric .... 


1.53 


Ether 




0.71 


Sulphuric . . 


1.85 


Glycerine . . 






1.26 


Acetone 


0.79 


Linseed Oil 






0.91 


Alcohol, absolute, 20° . 


0.789 


Olive Oil . . 






0.95 


90 per cent, 20° 


0.818 


Petroleum . . 






76-83 


Aldehvde 


0.81 


Turpentine 






0.87 


Benzine 


0.88 


Sea \Vater . . 




1.026 



4 


Reduction of Arbitrary Hydrometer Scales. 


Li 


jHter than Water. 


Heavier than Water. 


Sp. gr. 


B a ume. Beck. 


Cartier. 


Sp. gr. Baurne. 


Beck. Twaddell. 


0.75 


5SM 56°.7 




1.0 


O c .O 


o°.o 


O c .O 


0.80 


16.3 


12.5 


13.0 


1.1 


13.2 


15.1 


20.0 


0.85 


35.6 


30 .0 


33 . 6 


1.2 


21 .2 


28.3 


10.0 


0.90 


26.1 


18.9 


25.2 


1.3 


33.5 


39.2 


60.0 


0.95 


17.7 


8.9 


17.7 


1.1 


11.5 


18 .6 


80.0 


1.00 


10.0 


0.0 


11.0 


1.5 


48.4 


56 .7 


100.0 










1.6 


51.1 


63 .7 


120.0 










1.7 


59.8 


70 .0 


110.0 










1.8 


61 .5 


75.6 


160.0 










1.9 


68.6 




180.0 










2.0 


72 .6 




200 .0 



150 



PHYSICAL MEASUREMENT. 



5. Density of Aqueous Solutions. 



DENSITY. 



H 2 S0 4 



IIXO3 



II CI 



CuS0 4 



ZnSO. 



PbAc 2 


Sugar 


0.999 


0.999 


1.037 


1.019 


1.076 


1.039 


1.119 


1.060 


1.164 


1.082 


1.213 


1.105 


1.266 


1.129 


1.324 


1.153 


1.388 


1.178 




1.205 




1.232 




1.260 




1.289 




1.319 




1.349 




1.382 



Alcohol 
(Vol.) 





6 
10 
15 
20 

25 

30 
35 

40 

45 
50 
55 
60 

65 

70 
75 

80 

85 

90 

95 

1001 



5 

10 
15 
20 

25 
30 
35 
40 

45 

50 



10 
15 

20 

25 
30 
35 
40 

45 

50 



0.999 

1.033 
1.068 

1.105 
1.143 

1.182 
1.223 
1.264 
1.307 

1.352 
1.399 
1.449 
1.503 

1.558 
1.616 
1.676 
1.734 

1.786 
1.819 
1.839 

1.838 



NaOH 



0.999 

1.056 
1.111 
1.166 
1.222 

1.277 
1.333 

1.387 
1.442 

1.496 

1.548 



KOH 



0.999 

1.045 
L.092 
1.141 
1.191 

1.212 
1.295 
1.349 
1.406 

1.466 
1 .528 



0.999 

1.029 
1.05S 
1.089 
1.121 

1.154 

1.187 
1.220 
1.253 

1.287 
1.320 
1.350 
1.377 

1.402 
1.424 
1.443 
1.461 

1.479 
1.497 
1.514 
1.530 



0.999 

1.024 
1.049 
1.074 
1.100 

1.126 
1.152 
1.177 

1.200 



0.999 

1.050 
1.103 
1.161 

(1.225) 



0.999 

1.052 
1.108 
1.168 
1.236 

1.307 

1.382 



AgN0 3 

5% 1.043 
10" 1.090 
15 " 1.141 

20 " 1.197 



25' 
30' 
35' 
40' 

45' 
50' 
55' 
60' 



1.257 
1.323 
1.396 
1.479 

1.572 

1.677 
1.792 
1.919 



NaCl 



NaNO„ 



0.999 

1.035 
1.072 
1.110 
1.150 

1.191 



0.999 

1.032 
1.067 
1.103 
1.141 

1.181 
1.223 
1.267 
1.314 

1.365 
1.417 



KC1 



0.999 

1.032 
1.065 

1.099 
1.135 

(1.172) 



KXO3 



0.999 

1 .031 
1 .064 

1.099 
1.135 



Na 2 S0 4 



0.999 

1.045 
1.092 



K 2 S0 4 



0.999 

1 .040 
(1.083) 



K o C0 a 



0.999 

1.045 
1.092 
1.141 
1.192 

1.245 

1.300 
1.358 
1.417 

1.479 
1.543 



0.999 

0.992 
0.986 
0.980 
0.975 

0.970 
0.965 
0.958 
0.951 

0.943 
0.933 
0.923 
0.913 

0.901 

0.889 
0.876 
0.863 

0.849 
0.833 
0.815 
0.793 



JN a 2 VU3 


CaCl 2 


BaCl 2 


0.999 


0.999 


0.999 


1.052 


1.042 


1.045 


1.105 


1.086 


1.094 


(1.159) 


1.133 


1.148 




1.181 


1.205 




1.232 


1.269 




1.286 






1.343 






1.402 





MgS0 4 



0.999 

1.051 
1.105 
1.161 
1.221 

1.284 



K 2 Cr 2 7 


NH 3 


0.999 


0.999 


1.036 


0.978 


1.072 


0.958 


1.109 


0.941 




0.924 




0.910 




0.897 




0.885 



NELOl 



0.999 

1.015 
1.030 
1.044 
1.058 

1.073 



TABLES. 



151 



6. Density, D, of Water at 
Temperature, t°. 



t. 


D. 


DlFF. 


0° 

1 

2 
3 
4 
5 
6 
7 
8 
9 

10 

11 

12 

13 

11 

15 

16 

17 

18 

19 

20 

21 

22 

23 

21 

25 

26 

27 

28 

29 

30 


0.99988 
0.99993 
0.99997 
0.99999 
1.00000 
0.99999 
0.99997 
0.99991 
0.99988 
0.99982 
0.99971 
0.99965 
0.99955 
0.99913 
0.99930 
0.99915 
0.99900 
0.99881 
0.99866 
0.99817 
0.99827 
0.99806 
0.99785 
0.99762 
0.99738 
0.99711 
0.99689 
0.99662 
0.99635 
0.99607 
0.99579 


— 5 

— 1 

2 

— 1 
+ 1 

2 

3 

6 

6 

8 

9 

10 

12 

13 

15 

15 

16 

18 

19 

20 

21 

21 

23 

21 

21 

25 

27 

27 

28 

28 



Expansion of "Water from 
0° to 100°. 

Volume of 1 Gram of Water 
in Cubic Centimetres. 



Temp. 


Volume. 


IXC. PER 1°. 


0° 


1.0001 




4 


1.0000 




10 


1.0003 


0.00012 


15 


1.0009 


0.00016 


20 


1.0017 




25 


1.0029 


0.00021 
0.00028 


30 


1.0013 




35 


1.0059 


0.00032 
0.00036 


10 


1.0077 


0.00010 


45 


1.0097 




50 


1.0120 


0.00016 


55 


1.0111 


0.00018 


60 


1.0170 


0.00052 


65 


1.0197 


0.00054 


70 


1.0227 


0.00060 


75 


1.0258 


0.00062 


80 


1.0290 


0.00064 


85 


1.0323 


0.00066 


90 


1.0358 


0.00070 


95 


1.0395 


0.00074 


100 


1.0132 


0.00074 



8. Density of Gases. 



Gas. 


Density. 


At 0° temp, and 760 mm. 
pressure compared to 
^yater. 


Compared to air at sim- 
ilar pressure and tem- 
perature. 


Air 

Oxygen 

Xitrogen 

Hydrogen .... 
Carbonic dioxide . . 
Mixed gases from elec- 
trolysis of water 
Aqueous vapor . . . 


0.0012928 

0.0011293 

0.0012557 

0.00008951 

0.0019767 

0.0005361 


1.00000 
1.10563 
0.97137 
0.06926 
1.52910 

0.41472 
0.6230 



152 



PHYSICAL MEASUREMENT. 



9. Density of Dry Atmospheric Air at Temperature, t, and 
Barometric Pressure, b, reckoned from the Formula, — 

t, _ 0.001293 b 



1 + 0.00367 t 760 



Tem- 
pera- 
ture. 


Pbessube. 






5 = 700 


710 


720 


730 


740 


750 


760 


770 


P. P. 
17 


t 


0.00 


0.00 


0.00 


0.00 


0.00 


0.00 


0.00 


0.00 


0° 


1191 


1208 


1225 


1242 


1259 


1276 


1293 


1310 


mm. 
1 

2 
3 
4 
5 




1 


1187 


1204 


1221 


1237 


1254 


1271 


1288 


1305 


2 


o 


1182 


1199 


1216 


1233 


1250 


1267 


1284 


1301 


3 


3 


1178 


1195 


1212 


1228 


1245 


1262 


1279 


1296 


5 


4 


1174 


1191 


1207 


1224 


1241 


1258 


1274 


1291 


7 
8 


5 


1170 


1186 


1203 


1220 


1236 


1253 


1270 


1286 


6 


10 


6 


1165 


1182 


1199 


1215 


1232 


1249 


1265 


1282 


7 


12 


7 


1161 


1178 


1194 


1211 


1227 


1244 


1261 


1277 


8 


14 


8 


1157 


1174 


1190 


1207 


1223 


1240 


1256 


1273 


9 


15 


9 


1153 


1169 


1186 


1202 


1219 


1235 


1252 


1268 


lfi 


10 


1149 


1165 


1181 


1198 


1214 


1231 


1247 


1264 


mm. 




11 


1145 


1161 


1177 


1194 


1210 


1227 


1243 


1259 


1 


2 


12 


1141 


1157 


1173 


1190 


1206 


1222 


1238 


1255 


2 


3 


13 


1137 


1153 


1169 


1185 


1202 


1218 


1234 


1250 


3 


5 


11 


1133 


1149 


1165 


1181 


1198 


1214 


1230 


1246 


4 


6 


15 


1129 


1145 


1161 


1177 


1193 


1209 


1226 


1242 


5 


8 


16 


1125 


1141 


1157 


1173 


1189 


1205 


1221 


1237 


6 


10 


17 


1121 


1137 


1153 


1169 


1185 


1201 


1217 


1233 


7 


11 


18 


1117 


1133 


1149 


1165 


1181 


1197 


1213 


1229 


8 


13 


19 


1113 


1129 


1145 


1161 


1177 


1193 


1209 


1225 


9 


14 


20 


1110 


1125 


1141 


1157 


1173 


1189 


1205 


1220 


15 


21 


1106 


1122 


1137 


1153 


1169 


1185 


1200 


1216 


mm. 




22 


1102 


1118 


1133 


1149 


1165 


1181 


1196 


1212 


1 


1 


23 


1098 


1114 


1130 


1145 


1161 


1177 


1192 


1208 


2 


3 


24 


1095 


1110 


1126 


1141 


1157 


1173 


1188 


1204 


3 

4 

5 


4 


26 


1091 


1106 


1122 


1138 


1153 


1169 


1184 


1200 


6 

7 


26 


1087 


1103 


1118 


1134 


1149 


1165 


1180 


1196 


6 


9 


27 


1084 


1099 


1115 


1130 


1146 


1161 


1176 


1192 


7 


10 


28 


1080 


1095 


1111 


1126 


1142 


1157 


1173 


1188 


8 


12 


29 


1070 


1092 


1107 


1123 


1138 


1153 


1169 


11§4 


9 


13 


30 


1073 


1088 


1103 


1119 


1134 


1149 


1165 


1180 





TABLES. 



153 



10. Reduction of Volume of Gas to 0° C. a = 0.003665. 



t 


1+at 


t 


1+cct 


* 


1+at 


t 


1+at 


t 


1+at 


Prop. 
Parts. 


0° 


1.0000 


20° 


1.0733 


40° 


1.1466 


60° 


1.2199 


80° 


1.2932 




1 


1.0037 


21 


1.0770 


41 


1.1503 


61 


1.2236 


81 


1.2969 


16 


2 


1.0073 


22 


1.0806 


42 


1.1539 


62 


1.2272 


82 


1.3005 


1 


2 


3 


1.0110 


23 


1.0843 


43 


1.1576 


63 


1.2309 


83 


1.3042 


2 


3 


4 


1.0M7 


24 


1.0880 


44 


1.1613 


64 


1.2346 


84 


1.3079 


3 





5 


1.0183 


25 


1.0916 


45 


1.1649 


65 


1.2382 


85 


1.3115 


4 


6 


6 


1.0220 


26 


1.0953 


46 


1.1686 


66 


1.2419 


86 


1.3152 


5 


8 


7 


1.0257 


27 


1.0990 


47 


1.1723 


67 


1.2456 


87 


1.3189 


6 


10 


8 


1.0293 


28 


1.1026 


48 


1.1759 


68 


1.2492 


88 


1.3225 


7 


11 


9 
10° 


1.0330 
1.0366 


29 
30° 


1.1053 
1.1099 


49 
50° 


1.1796 
1.1832 


69 

70° 


1.2529 
1.2565 


89 

90° 


1.3262 
1.3298 


8 
9 


13 
14 


11 


1.0103 


31 


1.1136 


51 


1.1869 


71 


1.2602 


91 


1.3335 


17 


12 


1.0440 


32 


1.1173 


52 


1.1906 


72 


1.2639 


92 


1.3372 


1 


2 


13 


1.0476 


33 


1.1209 


53 


1.1942 


73 


1.2675 


93 


1.3408 


2 


3 


14 


1.0513 


34 


1.1246 


54 


1.1979 


74 


1.2712 


94 


1.3445 


3 


5 


15 


1.0550 


35 


1.1283 


55 


1.2016 


75 


1.2749 


95 


1.3482 


4 


7 


16 


1.0586 


36 


1.1319 


56 


1.2052 


76 


1.2785 


96 


1.3518 


5 


8 


17 


1.0623 


37 


1.1356 


57 


1.2089 


77 


1.2822 


97 


1.3555 


6 


10 


18 


1.0660 


38 


1.1393 


58 


1.2126 


78 


1.2859 


98 


1.3592 


7 


12 


19 


1.0696 


39 


1.1429 


59 


1.2162 


79 


1.2895 


99 


1.3628 


8 


14 


20° 


1.0733 


40° 


1.1466 


60° 


1.2199 


80° 


1.2932 


100° 


1.3665 


9 


15 



11. Reduction to Weight in Vacuo of the Weight of a Body 
of Density, D, weighed in Air with Brass Weights. 



D 


1c 


D 


Tc 


D 


ft 


D 


ft 


0.7 


+ 1.57 


1.6 


+ 0.61 


5 


+ 0.097 


14 


— 0.057 


0.8 


1.36 


1.7 


0.56 


6 


+ 0.057 


15 


— 0.063 


0.9 


1.19 


1.8 


0.52 


7 


+ 0.029 


16 


— 0.068 


1.0 


1.06 


1.9 


0.49 


8 


+ 0.007 


17 


— 0.072 


1.1 


0.95 


2.0 


0.46 


9 


— 0.009 


18 


— 0.076 


1.2 


0.86 


3.0 


0.26 


10 


— 0.023 


19 


— o.oso 


1.3 


0.78 


4.0 


0.16 


11 


— 0.034 


20 


— 0.083 


1.4 


0.71 


5.0 


0.10 


12 


— 0.043 


21 


— 0.086 


1.5 


0.66 






13 


— 0.050 







Reckoned from the formula, 



1000 



= 0,0012 [ - 

1 D 8.4 



If the weighed body has the density, D, and its weight in air be m 
grams, ink mg. must be added to reduce the weighing to vacuo. 



154 



PHYSICAL MEASUREMENT. 



12. Hygrometry Table. 

Pressure of aqueous vapor, p', and weight of water, h, contained in 1 cubic metre 
of air, with dew-point, t. 



t 


pi 


h 


t 


p' 


h 


t 


pi 


h 


t 


V 


h 




mm. 


g- 




mm. 


g- 




mm. 


g- 




mm. 


g- 


— 10° 


2.0 


2.1 


0° 


4.6 


4.9 


10° 


9.1 


9.4 


20° 


17.4 


17.2 


— 9 


2.2 


2.4 


1 


4.9 


5.2 


11 


9.8 


10.0 


21 


18.5 


18.2 


— 8 


2.4 


2.7 


2 


5.3 


5.6 


12 


10.4 


10.6 


22 


19.7 


19.3 


— 7 


2.6 


3.0 


3 


5.7 


6.0 


13 


11.1 


11.3 


23 


20.9 


20.4 


— 6 


2.8 


3.2 


4 


6.1 


6.4 


14 


11.9 


12.0 


24 


22.2 


21.5 


— 5 


3.1 


3.5 


5 


6.5 


6.8 


15 


12.7 


12.8 


25 


23.6 


22.9 


— 4 


3.3 


3.8 


6 


7.0 


7.3 


16 


13.5 


13.6 


26 


25.0 


24.2 


— 3 


3.6 


4.1 


7 


7.5 


7.7 


17 


14.4 


14.5 


27 


26.5 


25.6 


2 


3.9 


4.4 


8 


8.0 


8.1 


18 


15.4 


15.1 


28 


28.1 


27.0 


— 1 


4.2 


4.6 


9 


8.5 


8.8 


19 


16.3 


16.2 


29 


29.8 


28.6 


— 0° 


4.6 


4.9 


10° 


9.1 


9.4 


20° 


17.4 


17.2 


30° 


31.6 


30.1 



13. Capillary Depression of Mercury. 









Height of Meniscus 


IN MM. 






Diameter. 




















0.4 


0.6 


0.8 


1.0 


1.2 


1.4 


1.6 


1.8 


mm. 


mm. 


mm. 


mm. 


mm. 


mm. 


mm. 


mm. 


mm. 


4 


0.83 


1.22 


1.54 


1.98 


2.37 








5 


0.65 


0.86 


1.19 


1.45 


1.80 








6 


0.27 


0.41 


0.56 


0.78 


0.98 


1.21 


1.43 




7 


0.18 


0.28 


0.40 


0.53 


0.67 


0.82 


0.97 


1.13 


8 




0.20 


0.29 


0.38 


0.46 


0.56 


0.65 


0.77 


9 




0.15 


0.21 


0.28 


0.33 


0.40 


0.46 


0.52 


10 






0.15 


0.20 


0.25 


0.29 


0.33 


0.37 


11 




. 


0.10 


0.14 


0.18 


0.21 


0.24 


0.27 


12 






0.7 


0.10 


0.13 


0.15 


0.18 


0.19 


13 




• • 


0.4 


0.07 


0.10 


0.12 


0.13 


0.14 



TABLES. 



loo 



13 a. Barometer Correction for Temperature, t, for the Expan- 
sion of the Mercury and Brass Scale, the latter having 
a Coefficient, a — 0.000019. The Correction is to be sub- 
tracted. 



t 


680 


690 


700 


710 


720 


730 


740 


750 


760 


770 




mm. 


mm. 


mm. 


mm. 


mm. 


mm. 


mm. 


mm. 


mm. 


mm. 


1° 


0.11 


0.11 


0.11 


0.12 


0.12 


0.12 


0.12 


0.12 


0.12 


0.12 


2 


0.22 


0.22 


0.23 


0.23 


0.23 


0.24 


0.24 


0.24 


0.25 


0.25 


3 


0.33 


0.34 


0.34 


0.35 


0.35 


0.35 


0.36 


0.36 


0.37 


0.37 


4 


0.44 


0.45 


0.45 


0.46 


0.47 


0.47 


0.48 


0.49 


0.49 


0.50 


5 


0.55 


0.56 


0.57 


0.58 


0.58 


0.59 


0.60 


0.61 


0.62 


0.62 


6 


0.66 


0.67 


0.68 


0.69 


0.70 


0.71 


0.72 


0.73 


0.74 


0.75 


7 


0.77 


0.78 


0.79 


0.81 


0.S2 


0.83 


0.84 


0.85 


0.86 


0.87 


8 


0.88 


0.89 


0.91 


0.92 


0.93 


0.95 


0.96 


0.97 


0.98 


0.99 


9 


0.99 


1.01 


1.02 


1.04 


1.05 


1.06 


1.08 


1.09 


1.11 


1.12 


10 


1.10 


1.12 


1.13 


1.15 


1.17 


1.18 


1.20 


1.22 


1.23 


1.25 


11 


1.21 


1.23 


1.25 


1.27 


1.28 


1.30 


1.32 


1.34 


1.35 


1.37 


12 


1.32 


1.34 


1.36 


1.38 


1.40 


1.42 


1.44 


1.46 


1.48 


1.50 


13 


1.43 


1.45 


1.47 


1.50 


1.52 


1.54 


1.56 


1.58 


1.60 


1.62 


14 


1.51 


1.56 


1.59 


1.61 


1.63 


1.66 


1.68 


1.70 


1.72 


1.75 


15 


1.65 


1.68 


1.70 


1.73 


1.75 


1.77 


1.80 


1.82 


1.85 


1.87 


16 


1.76 


1.79 


1.81 


1.84 


1.87 


1.89 


1.92 


1.94 


1.97 


2.00 


17 


1.87 


1.90 


1.93 


1.96 


1.98 


2.01 


2.04 


2.07 


2.09 


2.12 


18 


1.98 


2.01 


2.04 


2.07 


2.10 


2.13 


2.16 


2.19 


2.22 


2.25 


19 


2.09 


2.12 


2.15 


2.19 


2.22 


2.25 


2.28 


2.31 


2.34 


2.37 


20 


2.20 


2.24 


2.27 


2.30 


2.33 


2.37 


2.40 


2.43 


2.46 


2.49 


21 


2.31 


2.35 


2.38 


2.42 


2.45 


2.48 


2.52 


2.55 


2.59 


2.62 


22 


2,42 


2.46 


2.49 


2.53 


2.57 


2.60 


2.64 


2.67 


2.71 


2.74 


23 


2^53 


2.57 


2.61 


2.65 


2.68 


2.72 


2.76 


2.79 


2.83 


2.87 


24 


2.64 


2.68 


2.72 


2.76 


2.80 


2.84 


2.88 


2.92 


2.95 


2.99 


25 


2.75 


2.79 


2.84 


2.88 


2.92 


2.96 


3.00 


3.04 


3.08 


3.12 


26 


2.86 


2.91 


2.95 


2.99 


3.03 


3.07 


3.12 


3.16 


3.20 


3.24 


27 


2.97 


3.02 


3.06 


3.11 


3.15 


3.19 


3.24 


3.28 


3.32 


3.37 


28 


3.08 


3.13 


3.18 


3.22 


3.27 


3.31 


3.36 


3.40 


3.45 


3.49 


29 


3.19 


3.24 


3.29 


3.34 


3.38 


3.43 


3.48 


3.52 


3.57 


3.62 


30 


3.30 


3.35 


3.40 


3.45 


3.50 


3.55 


3.60 


3.65 


3.69 


3.74 



156 



PHYSICAL MEASUBEMENT. 



14. Thermometer Correction for the Projecting Mercury- 
Thread. 

n = length in degrees of the projecting thread. 

t—t' = difference between the reading of the thermometer and the temperature 
of the air. 



t—t' 

n 
10 


30 

0.01 


35 


40 


45 


50 


55 


60 


65 


70 


75 


80 


85 


t— t' 


0.01 


0.05 


0.05. 


0.05 


0.06 


0.06 


0.07 


0.08 


0.09 


0.10 


0.10 


n 
10 


20 


0.12 


0.12 


0.13 


0.14 


0.15 


0.16 


0.17 


0.18 


0.19 


0.20 


0.22 


0.23 


20 


30 


0.21 


0.22 


0.23 


0.24 


0.25 


0.25 


0.27 


0.29 


0.31 


0.33 


0.35 


0.37 


30 


40 


0.28 


0.29 


0.31 


0.33 


0.35 


0.37 


0.39 


0.41 


0.43 


0.45 


0.48 


0.51 


40 


50 


0.36 


0.38 


0.40 


0.42 


0.44 


0.46 


0.48 


0.50 


0.53 


0.57 


0.61 


0.65 


50 


60 


0.15 


0.18 


0.51 


0.53 


0.55 


0.57 


0.60 


0.63 


0.66 


0.69 


0.73 


0.78 


60 


70 












0.66 


0.69 


0.71 


0.75 


0.81 


0.87 


0.92 


70 


80 














0.76 


0.81 


0.87 


0.93 


1.00 


1.06 


80 


90 


. 














0.92 


0.99 


1.06 


1.13 


1.20 


90 


100 


















1.10 


1.18 


1.26 


1.34 


100 



15. Boiling Temp, of Water, t, at Barometer Pressure, b. 



b 


t 


b 

1 


t 


b. 


t 


b 


t 


b 


t 


680 


96°.92 


700 


97°.72 


720 


98°.49 


740 


99°.26 


760 


100°.00 


681 


.96 


01 


.75 


21 


.53 


41 


.29 


61 


.04 


682 


97.00 


02 


.79 


22 


.57 


42 


.33 


62 


.07 


683 


.04 


03 


.83 


23 


.61 


43 


.37 


63 


.11 


684 


.08 


04 


.87 


24 


.65 


44 


.41 


64 


.15 


685 


.12 


05 


.91 


25 


.69 


45 


.44 


65 


.18 


686 


.16 


06 


.95 


26 


.72 


46 


.48 


m 


.22 


687 


.20 


07 


97.99 


27 


.76 


47 


.52 


67 


.26 


688 


.24 


08 


98.03 


28 


.80 


48 


.56 


68 


.29 


689 


.28 


09 


.07 


29 


.84 


49 


.59 


69 


.33 


690 


.32 


710 


.11 


730 


.88 


750 


.63 


770 


.36 


691 


.36 


11 


.15 


31 


.92 


51 


.67 


71 


.40 


692 


.40 


12 


.19 


32 


.95 


52 


.70 


72 


.44 


693 


.44 


13 


.22 


33 


98.99 


53 


.74 


73 


.47 


694 


.48 


14 


.26 


34 


99.03 


54 


.78 


74 


.51 


<m 


.52 


15 


.30 


35 


.07 


55 


.82 


75 


.55 


696 


.56 


1G 


.34 


36 


.11 


56 


.85 


76 


.58 


697 


.60 


17 


.38 


37 


.14 


57 


.89 


77 


.62 


698 


.64 


18 


.42 


38 


.18 


58 


.93 


78 


.65 


699 


.68 


19 


.46 


39 


.22 


59 


.96 


79 


.69 


700 


97°.72 


720 


98°.49 


740 


99°.26 


760 


100°.00 


780 


100°.72 



TABLES. 



157 



16. Melting-points and Boiling-points. 



Substance. 



Melting-point. 
Degree C. 


Boiling-point 
Degree C. 




56.53 




78.4 


About 700 




440 


1450 


268 


1450 


. 


90-110 


318 


768 


175 


204 


— 75 


97.2 


— 102 


— 33.6 


— 70 


61.2 


1500 




1100 




— 117.4 


34.9 




70-90 


20 


290 


1150 




114 


200 + 


1200 




1300-1400 




2200 




325 


1500 


180° 




750 


1100 


— 39 


357 


1450 




52-56 




44.2 


288 


2000 




97.6 


742 


( Melts 43 
j Solidifies 33 




{ Melts 47 
\ Solidifies 36 




62 





Acetone . . . 

Alcohol . . . 

Aluminum . . 

Antimony . . 

Bismuth . . . 

Benzine . . . 

Cadmium . . 

Camphor . . 

Chloral . . . 

Chlorine . . . 

Chloroform . . 

Cobalt. . . . 

Copper . . . 
Ether .... 

Gasoline . . . 

Glycerine . . 

Gold . . . . 
Iodine .... 

Iron, cast . . 

steel . . 

Iridium . . . 
Lead .... 

Lithium . . . 

Magnesium . . 

Mercury . . . 

Xickel . . . . 

Parafhne . . . 

Phosphorus . . 

Platinum . . 

Sodium . . . 

Tallow, beef . 

mutton 

Wax, bees . . 



158 



PHYSICAL MEASUREMENT. 



17. Critical Temperatures. 



Substance. 



Air 

Alcohol .... 
Acetone .... 
Carbon Dioxide 
Carbon Disulphide 

Ether 

Hydrochloric Acid 
Nitrogen .... 
Nitrous Oxide . . 
Oxygen .... 
Water 



Critical 
Tempera- 
ture. 



— 140 
234.3 
232.8 

31.1 

271.8 
190 
51.25 

— 146 

36.4 

— 118 
364 



Pressure 
in Atmos- 
pheres. 



39 
62.1 
52.2 
73 
74.7 
36.9 
86 
33 
73.1 
50 
194 






18. Latent Heats. 



Substance. 



Alcohol .... 
Bismuth .... 
Bromine .... 
Cadmium . . . 
Carbon Disulphide 

Ether 

Iodine 

Lead 

Methyl Alcohol . 
Mercury .... 
Phosphorus . . . 
Platinum .... 

Silver 

Sulphur .... 

Tin 

Turpentine . . . 

Water 

Zinc 



Heat of 
Fusion. 



12.64 

16.185 

13.66 



11.71 
5.85 

2.82 
4.7 

27.18 

21.07 
9.37 

13.31 

68 

79.24 

23 



Heat of 
Vaporiza- 
tion. 



208.92 



90 
90 



263.86 



536 



TABLES. 



159 



19. Specific Keats. 



Element. 



Aluminum 

Antimony 

Arsenic 

Bismuth 

Boron 

Bromine 

Cadmium . . . . . 

Calcium 

Carbon, diamond . . 

graphite 

charcoal . . 

Cobalt 

Copper 

Gold 

Iodine 

Iridium 

Iron 

Lead 

Lithium 

Magnesium . . . . 

Mercury, solid . . . 

liquid . . . 

Nickel 

Phosphorus, yellow 

red . . . 

Platinum 

Potassium 

Silicon ...... 

Sodium 

Sulphur, cryst. . . . 

Tin 

Zinc 

Substance. 

Beeswax, solid . . . 

liquid . . . 

Brass 

German Silver . . . 
Ice 

Alcohol 

Carbon Disulphide . . 
Chloroform . . . . 

Ether 

Paraffin e, solid . . . 

liquid . . . 

Turpentine . . . . 



Temperature. 


Specific Heat. 


to 100 


0.21S5 


to 100 


0.05 


21 to 68 


0.083 


20 to 81 


0.0305 


to 100 


0.2518 


13 to 15 


0.1071 


to 100 


0.056 


to 100 


0.1801 


15 to 1010 


0.366 


19 to 1010 


0.310 


to 221: 


0.2385 


9 to 97 


0.10671: 


to 100 


0.09332 


to 100 


0.032 


9 to 98 


0.0511 


to 100 


0.0323 


to 100 


0.113 


Oto 100 


0.0315 


27 to 99 


0.9108 


to 75 


0.2519 


— 78 to — 10 


0.03192 


to 100 


0.033 


11 to 97 


0.10916 


13 to 36 


0.202 


15 to 98 


0.1698 


Oto 100 


0.0326 


— 78.5 to 23 


0.1662 


57.1 


0.1833 


— 28 to 6 


0.2934 


17 to 15 


0.163 


to 100 


0.0559 


Oto 100 


0.0937 


20 to 58 


1.2 


65 to 100 


0.5 


15 to 98 


0.0888 


to 100 


0.09164 


— 20to0 


0.504 


16 to 30 


0.602 


30 


0.24 


Oto 8 


0.2354 


Oto 30 


0.535 


10 to 10 


0.589 


52 to 63 


0.708 


Oto 80 


0.46 



160 



PHYSICAL MEASUREMENT. 



20. Heats of Combustion. 

Calorics of heat liberated in the combustion of 1 kg. of the substance. 



Substance. 


PRODUCT 
OF 

Combustion. 


Heat of 
Combustion. 


Calcium 

Carbon, diamond . . . . 
graphite .... 

Coal 

Copper 

Gas, illuminating .... 

Hydrogen 

Magnesium 

Mercury 

Nitrogen 

Petroleum 

Phosphorus 

Silver 

Sulphur 

Turf 

Wood with 13# H 2 . . 

Zinc 


CaO 

C0 2 

co 2 

C0 2 
CuO 

H 2 

MgO 
(Hg 2 
IHgO 

p 2 o 

NO 

(no, 
p 2 o 5 

Ag 2 
S0 2 

ZnO 


3284 
7859 
7901 
8000 

590 

5600 

34500 

6045 

105 

103 

— 654 

— 1541 

— 143 
11000 

5747 
27 

2220 

4180 
( 4000 to 
)4400 

1300 



21. Coefficient of Cubical Expansion. 


Substance. 


Temperature. 


a = 3/? 


Alcohol 


to 78 


0.00104 


Ice 


— 20 to 1 


0.0001125 


Glass 


to 100 


0.00002664 


Mercury 


24 to 299 


0.0001815 


Paraffine 


16 to 38 


0.00039 


Quartz 


19 to 60 


0.0000353 


Rubber, soft .... 




0.000686 


Salt 


40 to 60 


0.000121 


Water 


to 100 


0.000062 






TABLES. 



161 



22. Coefficient of Linear Expansion. 


Substance. 


Tjempera- 

TUEE. 


? 


Aluminum 


40 

40 

40 

to 100 

40 to 50 

to 100 

52 
to 100 
to 100 
to 100 
40 
to 100 
to 100 
40 
40 
40 
2 to 34 

40 


0.00002313 

0.00001692 

0.00001152 

0.00001906 

0.00001792 

0.00001718 

0.00001934 

0.00001836 

0.0000088 

0.00001470 

0.000007 

0.000012 

0.000028 

0.00002694 

0.00001279 

0.00001921 

0.0000037 

0.0000584 

0.00000494 

0.0000544 

0.00002918 


, . r parallel to axis 

Antimony \ ,. . 

1 perpendicular to axis . . 

Brass, 71 Cu + 29 Zn 

Bronze, 86.3 Cu + 9.7 Sn + 4 Zn . . 

Copper 


Fluorite 


German Silver 


Glass 


Gold 

Iridium 


Iron 


Lead 

Magnesium 


Xickel 


Silver 


Wood 


Pine, with grain 

across grain 

Oak, with grain 

Zinc 





23. Thermal Conductivity. 

The heat which will pass through 1 c. mm. of a substance of absolute conductivity, 



C, in 1 second for a difference of temperature of 1° Trill heat C rag 
from 0° to 1°. To reduce to C. G. S. units, divide by 100. 



of water 



Substaxce. 



Aluminum . 
Antimony 

Bismuth . . 

Brass . . . 
Cadmium 

Copper . . . 
German Silver 

Glass . . . 
Ice .... 

Iron . . . 

Lead . . . 

Marble . . 

Magnesium . 

Mercury . . 

Silver . . . 



a 



SCBSTAXCE. 



c. 



34 

4 

1 

26 

22 

72 

8 





17 

8 

150 

37 

1 

109. 



.08 

.87 
.14 
.23 

.1 

.6 
.8 
6 



Alcohol . 
Ether . . 
Glycerine 
Petroleum 
Turpentine 
Water . . 



Air . . . 
Ammonia 
Hydrogen 
Nitrogen . 

Oxvgen . 



0.15 

0.37 

0.67 

0.035 

0.0325 

0.129 



0.0057 

0.006 

0.0327 

0.005 

0.0056 



1&2 



Pll YSICAL ME A S UEEMENT. 



24. Freezing Mixture. 






Substance. 


Pabts. 


Mixed with. 


Parts. 


Fall in 
Temp. 


Ammonium Nitrate . 


GO 


Water 


100 


27°.2 


Ammonium Chloride 


30 


Water . . . 






100 


18°.4 


Sodium Acetate . . 


85 


Water . . . 






100 


15°.4 


Sodium Hyposulphite 


110 


Water . . . 






100 


18°.7 


Alcohol at 4° ... 


77 


Snow . . . 






73 


30° 


Ammonium Chlorate 


25 


Snow at — 1° 






100 


15°.4 


Ammonium Nitrate . 


45 


Snow at — 1° 






100 


16°.7 


Sodium Nitrate . . 


50 


Snow at — 1° 






100 


17°.7 


Sodium Chloride . . 


33 


Snow at — 1° 






100 


21°.3 










Falls 










to 


Alcohol, absolute . . 




Carbon Dioxide, solic 




— 72° 


Chloroform .... 




Carbon Dioxide, solic 




— 77° 


Ether 




Carbon Dioxide, solic 




— 77° 


Methyl Chloride . . 




Carbon Dioxide, solic 




— 82° 


Sulphurous Acid . . 




Carbon Dioxide, solid 




— 82° 



25. Capillarity Constant {Surface Tension). 



Substance. 


Temperature. 


Surface Tension 

in Dynes per 

Linear cm. 


c 


10° 


24.97 


Alcohol -/ 


15° 

20° 


24.53 

24.09 


{ 


25° 


23.G5 


Chloroform 


20° 
10° 


27 
18.54 


f 


Ether J 


15° 

20° 


17.9G 
17.36 


| 


I 


25° 


16.79 


^lercury 


20° 


540 


r 


10° 


77.5 


Water 


15° 
20° 


76.6 




75.7 


. 


25° 


74.8 



TABLES. 



163 






26. Viscosity, or Coefficient of Internal Friction. 





Temp. 


1 


j Temp. 


n 


Acetone . 


20 


0.00331 


f 


10 


0.02977 


r 


12 


0.01182 




17 


0.01580 




20 


0.01257 


Mercury . J 


176 


0.01015 


Alcohol . X 


25 


0.01138 


1 


219 


0.00965 




30 


0.01034 


{ 


310 


0.00905 




50 


0.00715 


r 





0.0182 


Ether . . j 


12 
20 

25 


0.00278 
0.00258 
0.00215 


Water . . < 


10 
15 
17 

20 


0.0131 
0.01126 
0.01105 
0.0102 


f 


2.8 


42.20 




25 


0.0090 




8.1 


25.18 




30 


0.00788 


Glycerme J 


14.3 

20.3 

25.6 


13.87 
8.301 
5.113 


I 













27. Elasticity of 


Solids. 








Substance. 


i 

3 1 


Modulus 

OF 

Elasti- 
city, 


COEFF. 

of Elas- 
ticity, 


Limit 

OF 

Elasti- 
city, 


Breaking 
Stress, 


Modu- 
lus OF 
Tor- 
sion, 




H 


E 


1 
E 


L 


S 


T 




o 


Kg. 
sq. mm. 




Kg. 


Kg. 


Kg. 

sq. mm. 


sq. mm. 


sq. mm. 


Brass .... 


15 


9930 






60 


3550 


Bronze . . . 




9191 










Copper, wire . 


15 


12419 


0.00008 


12 


40 


J 3612 to 
I 4450 


German Silver 




12094 










Glass .... 


15 


6770 


. 






2567 


Gold, wire 




15 


8131 


0.000123 


. 


27 




Iron, cast 




15 


20794 


0.000018 


5 


48 




wire 




15 


20869 


0.000018 


32 


63 


6706 


Lead . . 




15 


1727 


0.000579 


0.225 


2.15 




Platinum 




15 


17044 




26 


34 




Silver . 




15 


7274 


0.000137 


11 


29 


2650 


Steel . . 




15 

15 


19519 
917 


0.000051 
0.001003 


33 
1.6 


83 


7900 


Wood — 

Birch . . 


With 

the 
Grain. 


Across 

the 
Grain. 


4.30 


0.82 


Maple 




15 


1021 


0.000979 


2.7 


2.71 


0.72 




Oak . 




15 


921 


0.001085 


2.3 


5.66 


0.58 




Pine . 




15 


1113 


0.00089 


2.2 


4.18 


0.22 




Poplar . 




15 


517 


0.001934 


1.5 


1.48 


0.14 




Zinc . . . 




15 


8734 


0.000114 









164 



PHYSICAL MEASUREMENT. 



28. Velocity of Sound in Metres per Second. 



SOLll>S. 


t 


V 


Liquids. 


t 


V 


Aluminum . . . 




5104 


Alcohol . . 


■i 


8°.4 


1264 


Brass . . 










3479 




23° 


1160 


Copper . . 








17° 


3825 


Ether . . . 




0° 


1150 


Glass. . . 








16° 


5150 


Petroleum . 




7°.4 


1395 


Gold . . . 








17° 


2081 


Turpentine . 




24° 


1212 


Iron . . . 








17° 


5000 




f 


3°.9 


1399 


Ivory . . 










3013 


Water . . . 


13°.7 


1437 


Magnesium 










4600 




25°.2 


1457 


Nickel . . 










4974 


Gases. 








Paraffine . 








16° 


1300 


Air .... 


•i 


0° 


332 


Silver . . 








17° 


2645 


20° 


344 


Thread, linen 








1815 


Chlorine . . 




0° 


206 


cotton 








1260 


Gas, ilium. . 




0° 


490 


Wood — 










Hydrogen 




0° 


1280 


Beech . 








3412 


Oxygen . . 




0° 


317 


Oak . . 








3381 


Vapors. 








Pine 








4650 


Alcohol . . 




48° 


236 








Ammonia 




0° 


415 








Ether . . . 




22° 


183 








Water . . . 




0° 
96° 


401 
410 



29. Pitch and Number of Vibrations per Second of 
Musical Notes. 





c- 2 


C-! 


C 


c 


Ci 


c 2 


c 3 


c 4 


c 


16.35 


32.70 


65.41 


130.8 


261.7 


523.3 


1047 


2093 


ct 


17.32 


34.65 


69.30 


138.6 


277.2 


554.4 


1109 


2218 


D 


18.35 


36.71 


73.42 


146.8 


293.7 


587.4 


1175 


2350 


Dt 


19.44 


38.89 


77.79 


155.6 


311.2 


622.3 


1245 


2489 


E 


20.60 


41.20 


82.41 


164.8 


329.7 


659.3 


1319 


2637 


F 


21.82 


43.65 


87.31 


174.6 


349.2 


698.5 


1397 


2794 


Ft 


2:5.12 


46.25 


92.50 


185.0 


370.0 


740.0 


1480 


2960 


G 


24.50 


49.00 


98.00 


196.0 


392.0 


784.0 


1568 


3136 


Gi 


25.96 


51.91 


103.8 


207.6 


415.3 


830.6 


1661 


3322 


A 


27.50 


55.00 


110.0 


220.0 


440.0 


880.0 


1760 


3520 


Ai 


29.13 


58.27 


116.5 


233.1 


466.2 


932.3 


1865 


3729 


B 


30.86 


61.73 


123.5 


246.9 


493.9 


987.7 


1975 


3951 



TABLES. 



165 



30. 



Reduction of Time of Oscillation to an Infinitely- 
Small Arc. 



k = 



l 



5 . a 

^i sm T' 
G4 4 



4 4 

If the observed time of oscillation of a magnet or pendnlum be t, with an arc of 
oscillation of a degrees, kt mnst be subtracted from the observed value in order 
to reduce the time to that of an infinitely small oscillation. 



a 


k 


a 


k 


a 


k 


a 


k 


0° 


0.00000 




10° 


0.00048 




20° 


0.00190 




30° 


0.00428 


29 
30 


1 


000 





11 


058 


10 


21 


210 


20 


31 


457 


2 


002 


2 


12 


069 


11 


22 


230 


20 


32 


487 


3 


004 


2 


13 


080 


11 


23 


251 


21 


33 


518 


31 


4 


008 


4 


14 


093 


13 


24 


274 


23 


34 


550 


32 
33 
33 
35 
35 
37 
38 


5 


012 


4 


15 


107 


14 


25 


297 


23 


35 


583 


6 


017 


5 


16 


122 


15 


26 


322 


25 


36 


616 


7 


023 


6 


17 


138 


16 


27 


347 


25 


37 


651 


8 


030 


7 


18 


154 


16 


28 


373 


26 


38 


686 


9 


039 


9 


IS 


172 


18 


29 


400 


27 


39 


723 


10° 


0.00048 


9 


20° 


0.00190 


18 


30° 


0.00428 


28 


40° 


0.00761 



10(5 



PII YSh \ 1 L MEASUREMENT. 



31. Lines of the Flame-Spectra of the Most Important 
Light Metals, 

according to Bunsen and KirchhofTs scale; the sodium line being taken 
as 50, and the slit having a breadth of 1 division. 

The first number denotes the position of the middle of the line 
upon the scale, the Roman numeral indicates the brightness, I being 
the brightest, and the third number gives the breadth of the band 
when it exceeds 1 scale division, the breadth of the slit. 

S signifies that the line is quite sharp and clearly defined, s that it 
is tolerably so; the remaining lines being nebulous and ill defined. 

The lines most characteristic of each body are printed in bold type. 

The brightness of the lines of C«, Sr, and Ba is that of a constant 
spectrum. If the chlorides be employed, the spectra are at first much 
brighter. In many cases the flame-spectra are really those of com- 
pounds, the spectra of the metals themselves obtained by the electric 
spark being frequently entirely different, and consisting of much finer 
lines. 

The colors of the spectrum are approximately, — red to 48, yellow 
to 52, green to 80, blue to 120, and violet beyond. 



K 


Na 


Li 


Ca 


Sr 


Ba 


17.5 II. s 




32.0 I. S 


33.1 IV. 2 
36.7 III. 


29.8 III. 
32.1 II. 
33.8 II. 




Faint con- 


50.0 I. S 


45.2 IV. s 


41.7 I. 1.5 

46.8 III. 2 


36.3 II. 
38.6*111. 


35.2 IV. 2 
41.5 III. 3 


tinuous 
spectrum 






49.0 III. 


41.5 III. 
45.8 I. 


45.6 III. s 1.5 


from 55 to 










52.1 IV. 


120. 






52.8 IV. 

54.9 IV. 
60.8 I. 1.5 
68.0 IV. 2 


105.0 III. s 


56.0 III. 2 
60.8 II. s 
66.5 III. 3 
71.4 III. 3 
76.8 III. 2 


153.0 IV. 






135.0 IV. S 




82.7 IV. 4 
89.3 III. 2 



TABLES. 



167 



2. Principal Bright Lines 


in Spe 


jctra of 


Elements. 




Symbol. 


Element. 


Wave- 
length. 


Scale 
No. 


Symbol. 


Element. 


Waye- 
Lexgth. 


Scale 

No. 






Ten-millionth 
mm. 








Ten-millionth 
mm. 






K« 


7680 


17.5 


b 


Mg 


5173 


76 




A 


7604 


18 


F 


H0 


4861 


90 




a 


7186 


23 




Sr<5 


4607. 


105 




B 


6870 


28.2 




Hy 


4340 


127 




Li a 


6708 


32.0 


G 




4307 


128 


C 


Ha 


6563 


34 


h 


H<5 


4102 


147 


D 


Na 


5892 


50 




KP 


4040 


153 




Tl 


5349 


68 


H 


Ca 


3966 


162 


E 


Ca 


5269 


71.3 


H' 


Ca 


3934 


166 



33. Wave-lengths of the Principal Fraunhofer Lines in 

o 

Angstrom's Units (ten-millionth mm.). 



Line. 


Wave- 
length. 


Source. 


Line. 


Wave- 
length. 


Source. 


A 


7594.059 





II 


3968.620 


Ca 


B 


6867.461 


o 


K 


3933.809 


Ca 


C 


6563.054 


H 


L 


3820.567 


Fe 


Dx 


5896.154 


Na 


M 


( 3727.763 
I 3727.20 


Fe 


D 2 


5890.182 


Na 


Fe 




f 5270.533 


Fe 


NT 


3581.344 


Fe 


E 


< 5270.448 


Ca 





3441.135 


Fe 




' 5269.722 


Ca 


P 


3361.30 


Fe 


bi 


5186.792 


Mg 


Q 


3286.87 


Fe 


b 2 


5172.871 


Mg 


It 


( 3181.40 
I 3179.45 


Ca 


b 3 


( 5169.218 


Fe 


Ca 




I 5169.066 


Fe 


Sx 


3100.779 


Fe 


b 4 


{ 5167.686 
i 5167.501 


Fe 


S 2 


3100.064 


Fe 




Mg 


T 


( 3021.191 
i 3020.759 


Fe 


F 


4861.496 


H 


Fe 


G 


( 4308.071 
I 4307.904 


Fe 


t 


2994.542 


Fe 




Ca 


u 


2947.993 


Fe 


h 


4101.870 


H 









168 



P1I YSICAL MEA S UIIEMENT. 



34. Color of Newton's Rings. 









Thickness 


of Layer. 


Order. 


Reflected. 


Transmitted. 


AIR. 


IODIDE OF 
SILVER. 








mm. 


mm. 




Black 


White . . . . 


0.000000 


0.0000000 




Lavender-gray 


Yellowish brown . 


0.000107 


0.0000116 




Bluish white . . 


Beddish brown . 


0.000124 


0.0000135 




Greenish white . 


Dark purple . . 


0.000129 


0.0000140 


I. 


Yellowish white . 


Dark violet . . . 


0.000135 


0.0000147 




Straw color . . . 


Dark blue . . . 


0.000140 


0.0000178 




Orpnge . . . . 


Bright blue . . . 


0.000235 


0.0000255 




Bed 


Pale blue-green . 


0.000245 


0.0000266 




Purple . . . . 


Pale green . . . 


0.000257 


0.0000279 




Violet 


Yellowish green . 


0.000272 


0.0000295 




Indigo . . . . 


Light yellow . . 


0.000282 


0.0000306 




Sky-blue .... 


Orange .... 


0.000352 


0.0000382 


IT. 


Light blue-green . 


Bed 


0.000372 


0.0000404 




Green 


Deep purple . . 


0.000387 


0.0000420 




Yellow . . . . 


Blue 


0.000435 


0.0000472 




Light orange . . 


Bright blue . . 


0.0004G5 


0.0000505 




Bed 


Bluish green . . 


0.000490 


0.0000532 




Purple .... 


Green 


0.000520 


0.0000565 




Violet .... 


Light yellow-green 


0.000550 


0.0000597 




Blue 


Yellow .... 


0.000570 


0.0000619 


III. 


Sea-green . . . 


Flesh color . . . 


0.000800 


0.0000652 




Green 


Purple .... 


0.000650 


0.0000706 




Greenish yellow . 


Gray-blue . . . 


0.000700 


0.0000755 




Bed 


Sea-green . . . 


0.000750 


0.0000814 




Purple .... 


Green .... 


0.000780 


0.0000847 




Gray-blue . . . 


Dull yellow . . 


0.000852 


0.0000925 


IV. 


Sea-green . . . 


Flesh color . . . 


0.000870 


0.0000945 




Green 


Grayish red . . 


0.000912 


0.0000990 




Bed 


Green .... 


0.000990 


0.0001082 


V. 


Blue-green . . . 


Flesh color . . . 


0.001108 


0.0001268 


Flesh color . . . 


Sea-green . . . 


0.001264 


0.0001373 


VI. 


Blue-green . . . 


Flesh color . . . 


0.001450 


0.0001574 



TABLES. 



169 



35. Index of Refraction — (for D line). 



Solids 



Agate . 
Alum 
Cadmium 
Copper . 
Diamond 
Fluor Spar 
Glass, flint 

crown 
Gold . . . 
Gypsum 
Ice . . . 
Iceland Spar 
Iron . . . 
Magnesium 
Mica . . . 
Nickel . . 
Opal . . . 
Platinum . 
Quartz . . 
Salt . . . 
Silver . . 
Sugar . . 
Sulphur 
Topaz . . 
Tourmaline 
Wax . . . 



AlR = 



1.54 
1.46 

1.13 

0.(34 

2.47 

1.46 

1.63 

1.52 

0.37 

1.52 

1.3 

1.58 

2.72 

0.37 

1.6 

1.87 

1.44 

1.64 

1.54 

1.54 

0.27 

1.56 

1.95 

1.63 

1.65 

1.5 



Liquids 



Alcohol . . . 
Benzine . . . 
Canada Balsam 
Carbon Disulphide 
Chloroform 
Ether . . . 
Olive Oil . . 
Turpentine 
Water . . . 



Gases. 



Air = l 



1.363 

1.5 

1.51 

1.63 

1.446 

1.36 

1.17 

1.18 

1.331 



Vacuum = 



Air 


1.000292 


Ammonia .... 


1.000376 


Carbon Dioxide . . 


1.000119 


Carbon Disulphide . 


1.001481 


Chlorine .... 


1.000773 


Ether 


1.001521 


Hydrogen .... 


1.000139 


Xitrogen .... 


1.000297 


Oxygen 


1.000272 


Water Vapor . . . 


1.000254 



36. Electromotive Force of Cells. 



Cell. 


E. Cell. 


E. 


Bmisen 

Clark, at 15° 

Daniell, copper sulphate 

and 9% sulphuric acid, 

Daniell, Gravity . . . 


1.68 
1.434 

1.078 
(about) 1 


Grenet (Chromic Acid), 

Grove 

LeClanche' 

Plante (Storage) . . . 
Water Element . . . 


1.8 

1.84 

1.4 

2.0 

1.0 



170 



PHYSICAL MEASUREMENT. 



37. Electrical Resistances. 

10 7 r = resistance of 1 cu. cm. = resistance of 1 sq. mm. x 1 Km, 

lOGOOr = specific resistance compared to mercury. 



Solids at 0°. 


10 7 r. 


Liquids at 18°. 


#• 


10 3 r. 


Aluminum . . . 


38 


Ammonia . . . 


NH 3 


1.7 


1123000 


Alum in. bronze . 


132 






8.8 


786000 


(90 Cu. + 10 Al.) 








18.2 


1887000 


Antimony . . . 
Bismuth .... 


377 
1084 


Copper Sulphate . 


CuS0 4 


5 

10 
15 


53300 
31450 
23890 


Brass 


(59 






17.5 


21940 


Cadmium . . . 


75 


Hydrochloric Acid 


HC1 


5 


' 2555 


Calcium .... 


75 






10 


1599 


Carbon .... 


59350 






20 


1323 


German Silver 


236 






30 
40 


1522 
1955 


Gold 


21 










Graphite . . . 


11500 


Nitric Acid . . . 


HN0 3 


6.2 
18.6 


3226 
1461 


Iron 


150 






31.0 


1289 


Lead 


196 






49.6 


1590 


Lithium .... 


88 






62 
Cone. 


2031 
6621 


Magnesium . . . 


43 










Mercury .... 


943 


Potassium Hydrate 


KOH 


5.36 


5492 


Nickel .... 


118 


Sodium Chloride . 


NaCl 


5 


15000 


Phosphor Bronze 


77 






10 
15 
20 


8333 
6146 
5154 


Platinum . . . 


135 






Sodium .... 
Silver 


51 
15 


Sodium Hydrate . 


NaOH 


4 
17 


6331 
2894 


Tin 


105 






30 


4965 


Zinc 


59 


Sulphuric Acid 


H 2 S0 4 


5 
10 
20 
30 
40 
50 
60 
70 
80 
85 
90 
99.4 
99.9 


4834 
2574 
1545 
1365 
1483 
1866 
2705 
4677 
9143 
10300 
9389 
118000 
7120 






Zinc Sulphate . . 


Zn S0 4 


5 
10 
20 
23.1 
25 


52700 
31340 
21490 
20420 
20960 



TABLES. 



171 



38. Temperature Coefficients of Electrical Conductivity. 

Conductivity, C t = C (1 — at). 



SUBSTAXCE. 


Temperature. 


a 


Aluminum 

Bismuth 

Brass 

Cadmium 

Carbon 

Copper 

German Silver .... 

Gold 

Graphite ...... 

Iron 

Lead 

Mercury 

Magnesium ...... 

Nickel 

Platinum , 

Silver 

Tin 

Zinc 


— 90 to 400 

to 100 

15 to 100 

12 to 100 

15 to 200 

to 100 

to 100 

12 to 100 

25 to 200 

Oto 100 

Oto 100 

— 50 to — 40 

Oto 100 

— 88 to 860 

Oto 100 
Oto 100 
to 100 
to 100 
100 to 360 


0.00388 
0.00440 
0.00125 
0.00:369 

— O.0003 
0.00394 
0.0003 
0.00367 

— 0.00075 
0.00531 
0.0039 
0.0028 
0.00098 
0.0039 
0.005 
0.003 
0.00384 
0.0051 
0.00419 • 



39. Specific Inductive Capacities. 

Air = 1. 



Solids. K. 


Liquids. 


K. 


Glass 

Gypsum ..... 

Ice 

Iceland Spar .... 

Marble 

Mica 

Paraffine 

Quartz 

Rosin 

Rubber, soft .... 
vulcanite . . 

Salt 

Sandstone 

Shellac 

Sulphur ..... 
Wood .,„... 


4to7 

5.6 

2.85 

7.4 

6.4 

6 to 8 

2.2 

4.54 

2.55 

2.4 

2.7 

5.8 

6.2 

3 

2.69 

2.95 


Acetone .... 
Alcohol .... 

Aldehyde . . . 
Benzine .... 
Carbon Disulphide 

Ether 

Glycerine . . . 

Oils 

Petroleum . . . 
Turpentine . . . 
Water .... 

Gases. 
Hydrogen . . . 
Vacuum .... 
Vapors .... 


21.8 

25 

18.6 

2.3 

2.5 

4.27 
56.2 

2.2 

2.06 

2.23 

75.5 

0.9998 
0.9985 
1.001 to 1.01 



172 



PHYSICAL MEASUREMENT. 



40. Electro-chemical Equivalents. 

Weight in mg. deposited by a current of 1 ampere. 





1'ik Sec. 


PBB Mix. 


Peb Hour. 


Copper 

Silver 

Water (II 2 0) . . . 


0.3281 

1.118 
0.0933 


19.(58 
67.09 
5.6 


1181 
4025 
336 



41. Magnetic Constants for the United States* 

The + sign denotes westerly declination. 



o 



o S 



5 * 

o g 



Alabama — 

Mobile 

Selma 

Arkansas — 

Little Rock . . . . 
California — 

Los Angeles . . . . 

San Francisco . . . . 
Colorado — 

Denver 

Connecticut — 

Hartford 

New Haven . . . . 
District of Columbia — 

Washington . . . . 
Florida — 

Jacksonville . . . . 
Georgia — 

Athens 

Atlanta 

Illinois — 

Lloomington . . . . 

Chicago ...... 

Evanston 

Galesburg 

Quincy 

Indiana — 

BloomingtoD - . . . 

Greencastle . . . . 



— 5.1 

— 4 

— 6.7 

— 14.2 

— 16.7 

— 14 

+ 10 
+ 10.8 

+ 5.9 

— 2 

— 2.5 



— D 

— 4.1 

— 4.1 

— 

— 6.7 

— 3.4 



+ 3.8 
+ 3.6 

+ 3 

+ 1 
— 0.5 

+ 3.1 

+ 3.5 
+ 3.8 

+ 2.4 

+ 3.5 

+ 3.7 
+ 3.6 

+ 3.8 
+ 3.8 
+ 3.8 
+ 3.9 
+ 3.9 

+ 3.5 
+ 3.5 



61 
63.2 

64.3 

59 
62.3 

67.7 

72.8 
72.5 

70.7 

61.6 

65 
64.8 

70.9 
72.2 
72.2 
71.3 
70.3 

70.2 
70.5 



Dynes. 
.2594 
.2650 

.2491 

.2712 
.2536 

.2271 

.1774 
.1814 

.2025 

.2716 

.2516 
.2536 

.2005 

.1880 
.1872 
.1952 
.2075 

.2097 
.2075 



• Tabulated from the charts of the United States ('oast and Geodetic Survey. 



TABLES. 173 

41. Magnetic Constants for the United States (Continued). 



j o 2 

< a ^j 

- " < 

z ? 5 



© 



hi 

H £ 

Z g 

C on 

h z 

£ ft 



Indiana {Continued) — 

Terre Haute 

Iowa — 

Des Moines 

Kansas — 

Lawrence — 9.4 

Kentucky — 

Louisville 

Louisiana — 

Baton Rouge — 5.9 

New Orleans - 5.6 

Maine — 

Brunswick ...... +15 

Maryland — 

Baltimore + 4.5 

Massachusetts — 

Boston +11.5 

Springfield + 10.5 

Michigan — 

Ann Arbor 

Minnesota — 

Minneapolis — 9 

Mississippi — 

Jackson — 6 

Missouri — 

Columbia — 7.5 

St. Louis — 6.2 

Montana — 

Helena — 19.5 

Nebraska — 

Lincoln — 10.5 

Nevada — 

Reno — 17 

New Hampshire — 

Concord +12.1 

Hanover + 12.9 

New Jersey — 

Princeton ! +7.3 



+ 3.5 

+ 4.5 

+ 3.4 

+ 3.6 

+ 4.2 
+ 4.2 

+ 2.4 

+ 3.1 

+ 3.6 
+ 3.3 

+ 3.1 

+ 3(?) 

+ 3.5 

+ 4 
+ 4-4 

+ 2(?) 

+ 4.6 

+ 2(?) 

+ 4.3 

+ 4 

+ 3.6 



70.3 

71.7 

68.7 

69.6 

60.1 
60 

74.7 

71.2 

73.5 
73.4 

73 

74.5 

02.5 

69.1 
69.3 

72.3 

70.3 

64.5 

74.1 
74 

71.9 



Dynes. 
.2075 

.1959 

.2190 

.2111 

.2746 
.2851 

.1614 

.1998 

.1728 
.1761 

.1824 

.1664 

.2646 

.2175 
.2175 

.1855 

.2075 

.2420 

.1656 
.1664 

.1888 



174 



PHYSICAL M Ei 18 U R EM h\\ T. 



41. Magnetic Constants for the United States (Concluded), 



H 
ft 



^ ° a 

g ! 5 

I S E 

fl - - 



hi 

h 

O cc 

§ S 

£5 



New York — 

Albany . . . 

New York . . 

Rochester . . 
North Carolina - 

Raleigh . . . 
North Dakota — 

Fargo .... 
Ohio — 

Athens . . . 

Cincinnati . . 

Columbus . . 
Oregon — 

Salem .... 
Pennsylvania — 

Allegheny . . 

Philadelphia . 
South Carolina — 

Charleston . . 
South Dakota — 

Vermilion . . 
Tennessee — 

Nashville . . 
Texas — 

Austin . . . 
Utah— 

Salt Lake City 
Vermont — 

Burlington . . 
Virginia — 

Charlottesville 

Lexington . . 
Washington — 

ttle . . . 
Wisconsin— 

Madison . . . 

Milwaukee . 



+ 9.9 
+ 8.3 
+ 6.7 

+ 1 

— 11 



— 2 

— 0.5 

— 20.6 

+ 3.5 
+ 6 

— 0.4 

— 10.8 

— 4 

— 8.8 

— 1G.7 

+ 12.5 

+ 3.4 
+ 2.4 

— 22 

— 6 

— 4.2 



+ 3 
+ 3.8 
+ 4.5 

+ 3.2 

+ 3 

+ 3.4 
+ 3.3 
+ 3.3 

+ 1 

+ 3.8 
+ 4.4 

+ 3 

+ 4(?) 

+ 4.6 

+ 3.6 

+ 2.5 

+ 5 

+ 2.5 
+ 3.4 

0(?) 

+ 4.8 
+ 5.4 



74 

72.4 

74.1 

67.9 

74.5 

70.6 
70.1 
70.9 

68.5 

72.2 
71.7 

64.1 

72.4 

67.1 

59 

67.1 

74.9 

69.7 
69.4 

71.1 

73.3 
73.4 



Dynes. 

.1729 
.1852 
.1658 

.2305 

.1724 

.2055 
.2075 
.1924 

.2092 

.1907 
.1938 

.2548 

.1924 

.2325 

.2816 

.2314 

.1660 

.2081 
.2135 

.1938 

.1800 
.1764 



TABLES. 



175 



CONSTANTS. 



Base of natural (Xaperian) logarithms, 
Modulus of natural logarithm, 
Log e, modulus of commom logarithms, 
Circumference of circle in degrees 
Circumference of circle in minutes 
Circumference of circle in seconds 
Circumference of circle, diameter unity, 



42. Numerical Constants. 

Number. 

e = 2.7182818 
= 2.3025851 
M = 0.4342945 
= 360 

= 21600 
= 1296000 
tt = 3.14159265 



Number. 

2tt= 6.2831853 
7T = 1.0471976 
3 



Logarithm. 
0.7981799 
0.0200286 



= 0.3183099 9.5028501-10 



tt2= 9.8696044 0.9942997 
The arc of a circle equal to its radius is 

In degrees, p° = 180/7r 

In minutes, p' = 60 p° 

In seconds, p" = 60 p' 
For a circle of unit radius, the 

Arc of 1° = l/p° 

Arc of V = \\p' 

Arc (or sine) of 1" = \j p" 



1/tt2 = 0.1013212 

V^ = 1.7724539 
1 

V2 = 1.4142136 
V3 = 1.7320508 



Logarithm. 
0.4342945 
0.3622157 
9.6377843-10 
2.5563025 
4.3344538 
6.1126050 
0.4971499 



10 



9.0057003 
0.2485749 
0.5641896 9.7514251-10 

0.1505150 
0.2385607 



= 57°.29578 
= 3437.7468' 
= 206264.S // 



= 0.0174533 
= 0.0002909 
= 0.00000485 



1.7581226 
3.5362739 
5.3144251 

8.2418774-10 
6.4637261-10 
4.6855749-10 



43. Geodetical Constants. 

Mean density of the earth = 5.527 

Surface density of the earth = 2.56 ± 0.16 

Acceleration of gravity : — 

^(cm. per second) = 980.60 (1 — 0,002662 cos 2 <£) for latitude <£ 

and sea level. 
(7, at equator = 977.99; g, at Washington = 980.07; g, at Paris 

= 980.94. 
g, at poles = 983.21; gr, at Greenwich = 9S1.17. 
Length of the seconds pendulum : — 

I = 39.012540 + 0.208268 sin- <£ inches = 0.990910 + 0.005290 sin 2 
<j> metres. 



176 PHYSICAL MEASUREMENT. 

44. Astronomical Constants. 

Sidereal year — 365.2563578 mean solar days. 
Tropica] year = 365.2422 days. 

Sidereal day 23 h. 56 m. 4.100 s. mean solar time. 
Mean solar day 24 h. 3 m. 50.540 s. sidereal time. 
Mean distance of the earth from the sun = 92800000 miles. 

45. Physical Constants. 

Velocity of light j in air = **»» km. per second. 
( in vacuo = 299940 km. per second. 
Velocity of sound through dry air, 332 m. per sec. 
Weight of distilled water, free from air, barometer 30 in., = 760 mm. 

Weight in Grains. Weight in Grams. 
Volume. 62° F. 4° C. 62° F. 4° C. 

1 cubic inch. 252.286 252.568 16.3479 16.3662 

1 cubic centimetre. 15.3953 15.4125 0.9976 0.9987 

1 cubic foot at 62° F. 62.2786 lbs. 

A standard atmosphere is the pressure of a vertical column of pure 
mercury whose height is 760 mm. and temperature 0° C, under 
standard gravity at latitude 45° and at sea level. 
1 standard atmosphere = 1033 grams per sq. cm. = 14.7 pounds 

per sq. in. 
Pressure of mercurial column 1 inch high = 34.5 grams per sq. 
cm. — 0.491 pounds per sq. in. 
Weight of dry air (containing 0.0004 of its weight of carbonic acid) : — 
1 cubic centimetre at temperature 32° F. and pressure 760 mm. and 
under the standard value of gravity weighs 0.00129305 gram. 
Specific heat of dry air compared with an equal weight of water: — 
At constant pressure, K p = 0.2374 (from 0° to 100° C, Kegnault). 
At constant volume, K v = 0.1689. 
Ratio of the two specific heats of air (Rontgen): K p /K v = 1.4053. 
Absolute zero of temperature, — 273°.0 C. = — 459°. 4 F. 
Mechanical equivalent of heat: — 

1 pound-degree, F. (the British thermal unit), = about 778 foot- 
pounds. 
1 pound-degree, C, = 1400 foot-pounds. 

1 calorie or kilogram-degree, C, = 3087 foot-pounds = 426.8 kilo- 
gram-metres = 4187 joules (for g = 9S1 cm.). 
Standard candle (sperm) burns 7.78 grams per hour. The flame is 
44.5 mm. high. 






TABLES. 



177 



UNITS. 

46. Synoptic Conversion of English to Metric Units. 

Units of Length. Metric Equivalents. Logarithms. 

1 inch. 2.54000 centimetres. 0.404835 

1 foot. 0.304801 metre. 9.4S4016 — 10 

1 yard. 0.914402 metre 9.961137-10 

1 mile. 1.60935 kilometre. 0.206650 



Units of Area. 
1 square inch. 
1 square foot. 
1 square yard. 
1 acre. 

1 square mile. 
1 square mile. 



6.4516 cquare centimetres. 0.809669 

929.034 square centimetres. 2.968032 

0.83613 square metre. 9.922274-10 

0.404687 hectare. 9.607120 — 10 

2.5900 square kilometres. 0.413300 

259 hectares. 2.413300 



Units of Volume. 
1 cubic inch. 
1 cubic foot. 
1 cubic yard. 



16.3872 cubic centimetres. 1.214504 

0.028317 cubic metre or stere. 8.452047 — 10 

0.76456 cubic metre or stere. 9.883411—10 



Units of Capacity. 

1 gallon (U. S.) = 231 cubic inches. 

1 quart (U.S.). 

1 Imperial gallon (British). 

277.463 cubic inches (1890). 
1 bushel (U. S.)= 2150.42 cubic inches. 35.2393 litres. 1.547027 



3.78544 litres. 
0.94636 litre. 
4.5468 litres. 



1 bushel (British). 

Units of Mass. 
1 grain. 

1 pound avoirdupois. 
1 ounce avoirdupois. 
1 ounce troy, 
lton (2240 lbs.). 



64.7989 milligrams. 

0.4535924 kilogram. 
28.3495 grams. 
31.1035 grams. 

1.01605 tonne. 



0.578116 

9.976056- 

0.657709 



-10 



36.3477 litres. 1.560477 



1.811568 

9.656666- 

1.452546 

1.492809 

0.006914 



-10 



Units of Velocity. 

1 foot per sec. (0.6818 mile per hr.)= 0.30480 metre per sec. = 1.0973 

km. per hr. 
1 mile per hr. (1.46667 foot per sec.)= 0.44704 metre per sec. = 1.6093 

km. per hr. 



ITS 



PHYSICAL MEA8UBEMENT. 



Synoptic Conversion of English to Metric Units. — Continued. 

Units of Force. 
1 poundal 13826.5 dynes. 4.140682 

Weight of 1 grain (for g = 981 cm.). 63.57 dynes. 1.803237 

Weight oi 1 pound av.(for g =981 cm.). 4.45 x 10 5 dynes. 5.648335 

Units of Stress — /?? Gravitation Measure. 

1 pound per square inch = 70.307 grains per sq. cm. 
1 pound per square foot = 4.8824 kilograms per sq. m. 

Units of "Work — in Absolute Measure. 
1 foot-poimdal 421403 ergs. 

Units of "Work — in Gravitation Measure. 
1 foot-pound (for g = 981 cm.) = 185(3.3 x 10 4 ergs = 0.138255 kilo- 
gram-metre. 

Units of Activity (rate of doing work). 
1 foot-pound per minute (for g = 981 cm.) = 0.022G05 watt. 
1 horse-power (33000 foot-pounds per min.) = 746 watts = 1.01387 
force de ckeval. 



1.846997 
0.688634 



5.624697 



Units of Heat. 
1 pound-degree, F. 
1 pound-degree, C. 



252 small calories or gram-degrees, C. 
1.8 pound-degrees, F. 






47. Synoptic Conversion of Metric to English Units. 
Units of Length. English Equivalents. Logarithms. 

1 metre (10* microns). 39.3700 inches. 1.595165 

1 metre. 3.28083 feet. 0.515984 

1 metre. 1.09361 yard. 0.038863 

1 kilometre. 0.62137 mile. 9.793350-10 



Units of Area. 
1 square centimetre. 
l Bquare metre. 
l Bquare metre. 
1 hectare. 
1 square kilometre. 

Units of Volume. 

l cubic centimetre. 

1 cubic metre or stere. 
1 cubic metre or si 



0.15500 


square inch. 


9.190331- 


-10 


10.7039 


square feet. 


1.031968 




1.19599 


square yard. 


0.077726 




2.47104 


acres. 


0.392880 




0.38610 


square mile. 


9.586700- 


-10 


0.0610234 cubic inch. 


8.785490- 


-10 


35.3145 


cubic feet. 


1.547953 




1.30794 


cubic yard. 


0.116589 





TABLES. 



179 



Synoptic Conversion of Metric to English Units. — Continued, 

Units of Capacity. 
1 litre (61.023 cubic inches). 0.26417 gallon (U. S.). 



10 



1 litre. 
1 litre. 
1 hectolitre. 
1 hectolitre. 

Units of Mass. 
1 gram. 
1 kilogram. 
1 kilogram. 
1 kilogram. 
1 tonne. 

Units of Velocity. 
1 metre per second. 
1 metre per second. 



1.05668 quart (U.S.). 
0.21993 Imp. gal. (British). 
2.83774 bushels (U.S.). 
2.7512 bushels (British). 



9.421884- 
0.023944 
9.342291-10 
0.452973 

0.439523 

1.188432 



15.4324 grains. 

2.20462 pounds avoirdupois. 0.343334 

35.274 ounces avoirdupois. 1.547454 



32.1507 ounces troy. 
0.98421 ton (2240 lbs.). 

3.2808 feet per second. 
2.2369 miles per horn*. 



1.507191 

9.993086- 



-10 



0.515984 
0.349653 
9.793350-10 



lkm.perh.(0.277Sm.persec). 0.62137 mile per hour. 

Units of Force. 

1 dyne (weight of (981)- 1 grams, for g = 981 cm.) = 7.2330 x 10~ 5 
poundals. 

Units of Stress — in Gravitation Measure. 

1 gram per square centimetre. 0.014223 pound per sq. inch. 

1 kilogram per square metre. 0.20482 pound per sq. foot. 

1 standard atmosphere. 14.7 pounds per sq. inch. 

Units of "Work — in Absolute Measure. 
1 erg. 2.3730 X 10~ 6 foot-poundals. 

1 megalerg == 10 8 ergs; 1 joule = 10" ergs. 

Units of "Work — in Gravitation Measure. 
1 kilogram metre (for #=981 cm.)=981xl0 5 ergs = 7.2330 foot- 
pounds. 
Units of Activity (rate of doing work). 
1 watt. 44.2385 foot-pounds per minute, for g = 981 cm. 

1 watt = 1 joule per sec. = 0.10194 kilogram-metres per sec, 
for g = 981 cm. 
1 force de cheval = 75 kilogram-metres per sec. = 735| watts = 
0.98632 horse-power. 
Units of Heat. 
1 calorie or kilogram-degree = 3.968 pound-degrees, F. = 2.2046 

pound-degrees, C. 
1 small calorie or therm, or gram-degree = 0.001 calorie or kilogram- 
degree. 



180 



PHYSICAL MEASUREMENT. 



48. 

Quantity. 

Area. 
Volume. 

Mass. 

Density. 

Velocity. 

Acceleration. 
Angle. 

Angular Velocity. 



Dimensions of Physical Quantities. 

L = length ; M = mass ; T = time. 



Dimensions. Quantity. 

[L 2 ] Momentum. 

[L 3 ] Moment of Inertia 

[M] Force. 

[ML -3 ] Stress (per unit area). 

[LT- 1 ] Work or Energy. 

[LT- 2 ] Rate of Working. 

[0] Heat. 

[T- 1 ] Thermal Conductivity. 



e 
cr 



F 

C or q 

k 



In Electrostatics. Symbol. 

Quantity of Electricity. 

Surface Density: quantity per unit area. 

Difference of Potential : quantity of work required 
to move a quantity of electricity; (work done)-|- 
( quantity moved). 

Electric Force, or Electromotive Intensity: (quan- 
tity) -f- (distance 2 ). 

Capacity of an Accumulator: e -r E, 

Specific Inductive Capacity. 

In Magnetics. 

Quantity of Magnetism, or Strength of Pole. 

Strength or Intensity of Field: (quantity) -r (dis- 
tance 2 ). 

Magnetic Force. 

Magnetic Moment: (quantity) x (length). 

Intensity of Magnetization: magnetic moment per 
unit volume. 

Magnetic Potential : Work done in moving a quan- 
tity of magnetism; (work done) -f- (quantity 
moved). 

Magnetic Inductive Capacity (permeability). 



Dimensions. 
[LMT- 1 ] 

[ML 2 ] 

[LMT- 2 ] 

[L-iMT- 2 ] 

[L 2 MT- 2 ] 

[L 2 MT~ 3 ] 

[L 2 MT~ 2 ] 

[L-iMT- 1 ] 

Dimensions in 

Electrostatic 

System. 

[L^M^T- 1 ] 

[L-*M*T _1 ] 



E [L*M*T _1 ] 



8 

m 

ml 



[irhiiT- 1 ] 
[L] 

[0] 

Dimensions in 
Electro-Magnetic 
System. 

[LiM^T" -1 ] 

[L-^M'T" 1 ] 
[L^M^T- 1 ] 
[LtM*T _1 ] 



[L- 



_ *M*T~ 



*] 



For O 



[L^T" 1 ] 
[0] 



In Electro-Magnetics. 

Intensity of Current. 

Quantity of Electricity conveyed by 

Current: (intensity) x(time). 



Dimensions in 
Symbol. Electro-Magnetic 

System. 

i [L*M*T~ l ] Ampere. 



Name of 

Practical 

Unit. 



[L*M*] 



Coulomb. 



TABLES. 181 

Dimensions of Physical Quantities. — Continued. 

Potential, or Difference of Potential: 
(work done) -^(quantity of electri- 
city upon which work is done), E [iJM^T - ' 2 ] Volt. 

Electrical Force: the mechanical 
force acting on electro-magnetic 
unit of quantity: (mechanical 
force )-f( quantity). E [L*M*T~ 2 ] 

Resistance of a Conductor: E -=- i. B [LT- 1 ] Ohm. 

Capacity: quantity of electricity 
stored up per unit potential differ- 
ence produced by it. q [L^T 2 ] Farad. 

Specific Conductivity: the intensity 
of current passing across unit area 
under the action of unit electric 
force. [L~ 2 T] 

Specific Resistance : the reciprocal of 

specific conductivity. r [L-T - x ] 



Use of the Logarithm Table. 

A common or Briggs logarithm is merely the expression of a num- 
ber as a power of 10. Thus, 724 = 10-- 8597 . 2.S59T is called the log. of 
724. From the theory of exponents we get the following rules for 
calculating with logarithms : — 

(1) To multiply two numbers, add their logs., and the resulting 
sum will be the log. of their product. 

(2) To divide one number by another, subtract the log. of the sec- 
ond from that of the first, and the result will be the log. of the quotient. 

(3) To raise a number to any power, multiply its log. by the expo- 
nent of that power, and the result is the log. of the required power. 

(4) To extract any root of a number, divide its log. by the index 
of the root, and the result will be the log. of the required root. 

Expressed as formulae, these rules become, — 

(1) log a- b = log a + log b. (2) log a/b = log a — log b. 
(3) log a h = b log a. (4) log Va = (log a) lb. 

The logarithm 2.8597 is seen to be divided into two parts, a whole 
number, 2, and a fraction, .S597. The integral part of a logarithm is 
called its characteristic, and its size depends merely on the position 
of the decimal point. The fractional part is called the mantissa. The 
mantissa is always positive. In the table, the mantissas only are 
given, and the proper characteristic must be supplied by the student. 



182 PHYSICAL MEASUREMENT. 

All numbers which can be derived from each other by multiplying 
or dividing by integral powers of 10 have the same mantissa. 

(I.) The characteristic of a number greater than one is positive, and 

one unit less than the number of places to the left of the decimal point. 

(II.) The characteristic of a number less than one is negative, 
and one unit more than the number of zeros before the first significant 
figure o( the number expressed as a decimal fraction. 

(III.) If the characteristic is positive, make the number of figures 
before the decimal point one more than the number of units in the 
characteristic. 

(IV.) If the characteristic is negative, make the number of zeros 
between the decimal point and the first significant figure of the cor- 
responding fraction, one less than the number of units in the 
characteristic. 

Thus, the logarithm of 724 is 2.8597; of 72.4, 1.8597; of 7.24, 
0.S597; of 0.724, 1.8597; of 0.0724, 2.8597. The logarithm 2.8597 is 
the same as .8597 — 2, or, 8.8597 — 10. 

It is customary to write negative characteristics in this way for con- 
venience in calculation, 10 or some multiple of 10 being subtracted from 
the logarithm, and a number written before the decimal point, which, 
when 10 is subtracted from it, equals the real negative characteristic. 

Thus, 3.3496 is written 7.3496 - 10; 5.3496, 5.3496 — 10; 12.3496, 
8.3496 - 20. 

TO FIND THE LOG. OF A NUMBER. 

For example, to find the logarithm of 478. Look down the column 
of numbers at the left, until the first two figures (47) are found; then 
look along the horizontal line of logarithms until the vertical column 
headed by the third figure (8) is reached. Here the desired logarithm 
is found to be 2.6794, the characteristic being supplied according to 
the above rules. In the same way, the logarithm of .352 is found to be 
L.5465. If the number contains more than three figures, the exact 
logarithm cannot be found in the table, and interpolation must be 
used. This is based on the assumption that the change between two 
consecutive mantissas in the table is proportional to the change in the 
numbers. For example, to find the logarithm of 3.464. The logarithm 
of 3.46 is .5391. The difference between the logarithms of 3.46 and 3.47 
n to be .0012. Xow, the extra figure is 4; hence, we must add to 
the logarithm of 3.46, T 4 - of the difference, .0012, or .00048. The log- 
arithm of 3.464 is .5396, i^iven to four places. To simplify this process, 
in tie- columns to the right of the last heavy line are given the numbers 
to be added when the fourth figure is that at the head of the column. 
Thus, in the last ease we might have found the logarithm of 3.46 to be . 

1. and thru looked along the same horizontal line in the correction 
columns until the one headed four was reached; here the correction to 



TABLES. 183 

be added would be found to be .0005 (the zeros being left out), the 
sanie as before. In the same way, the logarithm of 89.82 is 1.9533 + 
.0001 = 1.9534. 

To find a number from its logarithm. 

If the given mantissa is found in the table, the first two figures 
will be on the same horizontal line in the column at the left, and the 
third at the top of the page, in the same vertical column with the man- 
tissa, the decimal point being determined from the characteristic. Thus, 
the number corresponding to 2.5211 is 332. If the exact mantissa is not 
found in the table, note the nearest smaller mantissa found there, and 
the first three figures will be those corresponding to this. Then note 
the difference between this and the given mantissa, and look along the 
same line in the correction table, until this difference, or the nearest 
difference to it is seen; and at the head of this column, the fourth 
figure will be found. Thus, to find the number corresponding to 2.6519. 
The nearest smaller mantissa in the table is .6513; then the first three 
figures are 44S. The difference between .6519 and .6513 is .0006, and 
the number of the head of the column in which 6 is found is 6, hence 
the whole number is 448.6. 

Example of the use and arrangement of logarithms. Suppose we 
have the equation to solve, — 

422- 0.3*. 0.17 _ A _ ^ 





^a 


14 


•0.277 
log' 


B 

122 = 2.6253 


)g0.3 


= 9.4771 - 10; 




4.log 
log 


0.3 = 7.6084-10 
0.17 = 9.2304 - 10 

19.7641 — 20 = log A. 


>g.l4 


= 29.1461 -30; 




log 


0.14= 9.7154-10 
0.277 = 9.4425 - 10 

19.1579 —20 = log B. 




log A — log B 


= 


.6062 = 


= logJ?; B = 4.038. 



L~SE OF THE TRIGONOMETRIC TABLES. 

In general, these tables are used in very much the same way as the 
logarithm tables. The degrees are given in the column at the left, and 
the functions are given directly in the table for each tenth of a degree, or 
every six minutes. Thus, in the table of sines, the sine of 11° is .1908; 
of 11° 24' is .1977. The columns of the differences at the right give 
the number to be added in the last place of figures of the function for 
any number of additional minutes between 1' and 5'. 

In this way, the functions can be found for each minute of arc. 
For example, the sine of 11° 26' is the sine of 11° 24' plus the difference 
corresponding to 2; or, .1977 + -0006 = .1983. 



184 



LOGAlUTiniS. 








1 


2 


3 


4 


5 


6 


7 


8 


9 


12 3 


4 5 


G 


7 8 9 


10 


oooo 


0043 






0170 


0212 


0253 


0201 


0334 


0374 


4 8 


12 


17 21 


25 


29 33 37 


11 

18 

13 


0114 
0792 

1139 


0453 

0S2S 
1173 


0492 
0661 

1200 


0531 
0899 
1239 


0569 
0934 

1271 


0607 
0969 

1303 


0645 

1(104 
1335 


0682 

1038 
1367 


0710 
1072 

1300 


0755 
1106 
1430 


4 8 
3 7 
3 6 


11 
10 

10 


15 19 23 
14 17 21 
13 16 19 


26 30 34 
24 28 31 
23 26 20 


11 
15 
lfi 


14G1 

1761 
2041 


1192 
171H) 
2068 


1523 
1818 

2095 


1553 
1817 
2122 


15S4 
1875 
2148 


1614 

1003 
2175 


1644 
1931 
2201 


1673 
1959 
2227 


1703 
1987 

2253, 


1732 
2014 

2270 


3 6 
3 6 
3 5 



8 

s 


12 15 18 
11 14 17 
11 13 16 


21 24 27 
20 22 25 
18 21 24 


17 
18 

10 


2301 
2553 

2788 


2330 
2577 
2810 


2355 
2601 
2833 


2380 
2625 
28 -.6 


2405 
2648 

2878 


2130 
2672 

2000 


2455 
2695 
2923 


2480 
2718 
2945 


2504 
2742 
2967 


2529 
2765 
2989 


2 5 
2 5 
2 4 


7 
7 
7 


10 12 15 
9 12 14 
9 11 13 


17 20 22 
16 19 21 
16 18 20 


80 


3010 


3032 


3054 


3075 


3096 


3118 


3139 


3160 


3181 


3201 


2 4 


6 


8 11 


13 


15 17 19 


81 

22 

2:5 


3222 
3124 
3G17 

3S02 
3079 
4150 


3213 
3444 
3636 

3820 
3997 
416G 


3263 
3464 

3655 


3284 
3183 
3671 


3304 
3502 

3(102 


3321 
3522 
3711 


3345 
3541 
3729 

3909 
4082 
4240 


3365 
3560 
3747 


3385 
3579 
3766 


3404 
3598 
3784 


2 4 
2 4 
2 4 


6 
6 

6 


8 10 12 
8 10 12 
7 9 11 


14 16 18 
14 15 17 
13 15 17 


24 
25 

26 


3838 
4014 
4183 


3856 
1031 
4200 


3874 
4048 
4216 


3892 
4065 
4232 


3927 
4099 
4265 


3945 
4116 
4281 


3962 
4133 
4298 


2 4 
2 3 
2 3 


5 

5 
5 


7 9 11 
7 9 10 

7 8 10 


12 14 16 
12 14 15 
11 13 15 


27 

28 
29 


4314 
4472 
4624 


4330 

4487 
4639 


4346 
4502 
4654 


4362 
4518 
4669 


4378 
4533 
4683 


4393 

4518 
4698 


4400 
4564 
4713 


4425 
4579 
4728 


4440 
4594 
4742 


4456 
4609 
4757 


2 3 
2 3 
1 3 


5 
5 
4 


6 8 
6 8 
6 7 


9 
9 

9 


11 13 14 
11 12 14 
10 12 13 


30 


4771 


4786 


4800 


4811 


4829 


4S13 


4857 

4997 
5132 
526.') 


4871 


4886 


4900 


1 3 


4 


6 7 


9 


10 11 13 


31 
32 

33 


4914 
5051 

5185 


4928 
5065 
5198 


4912 
5079 
5211 


1955 
5092 
5221 


4969 
5105 
5237 


49S3 
5119 
5250 


5011 
5145 
5276 


5024 
5159 

5289 


5038 
5172 
5302 


1 3 
1 3 
1 3 


4 
4 

4 


6 7 
5 7 
5 6 


8 
8 
8 


10 11 12 
9 11 12 
9 10 12 


31 
35 
30 


5315 
5441 

5503 


5328 
5153 

5575 


5310 
5165 

5587 


5353 

5178 
5599 


5366 
5490 
5611 


5378 
5502 

5623 


5391 
5514 
5635 

5752 
5866 
5977 


5403 
5527 
5647 

5763 

5877 
5988 


5416 
5539 
5658 


5428 
5551 
5670 


1 3 
1 2 
1 2 


4 
4 
4 


5 6 
5 6 
5 6 


8 
7 
7 


9 10 11 
9 10 11 
8 10 11 


37 
38 
39 


56S2 
5798 
5911 


5691 
5809 
5922 


5705 
5821 
5933 


5717 
5832 
5944 


5729 
5843 
5955 


5740 
5855 
5966 


5775 
5888 
5999 


5786 
5899 
6010 


1 2 
1 2 
1 2 


3 
3 
3 


5 6 

5 6 
4 5 


7 

7 
7 


8 9 10 
8 9 10 
8 9 10 


40 


0021 


6031 


6012 


6053 


6004 


6075 


6085 


6096 


6107 


6117 


1 2 


3 


4 5 


6 


8 9 10 


41 
42 

43 


G128 
6232 
6335 


6138 
6213 
6315 


6149 
6253 
6355 


6160 
6263 
6365 


6170 
6274 

6:575 


6180 
6284 
6385 


6191 
6294 
6305 


6201 
6304 
6405 


6212 
6314 

6415 


6222 
6325 
6425 


1 2 
1 2 
1 2 


3 
3 

3 


4 5 
4 5 
4 5 


6 
6 
6 


7 8 9 
7 8 9 

7 8 9 


14 
45 
16 

17 
18 

19 


0135 
6532 


6444 
6512 
6637 


6454 
6551 
6616 


6461 
6561 
6656 

6749 
6839 

Ton; 

7101 

7185 


6471 

6571 
6665 


6481 
6580 
6675 


6493 
6590 

6684 


6503 
6599 
6693 


6513 
6609 

0702 


6522 
6618 
6712 


1 2 
1 2 
1 2 


3 
3 
3 


4 5 

4 5 

4 5 


6 

6 



7 8 9 
7 8 9 

7 7 8 


G721 
6812 
6902 

71G0 
7243 


6730 
6821 

6911 

6998 


6830 
6920 


6758 

6818 
6937 


6767 
6857 
6916 


6776 
6866 
6055 

7042 


6785 

6875 
6964 


6794 
6884 
6972 


6803 
6S93 
6981 


1 2 

1 2 
1 2 


3 
3 
3 


4 5 
4 4 

4 4 


5 
5 
5 


6 7 8 

6 7 8 
6 7 8 


go 


7007 


7021 


7033 

7118 
7202 

7284 


7050 


7059 


7067 


1 2 


3 


3 4 


5 


6 7 8 


51 
59 

58 


7081 

7168 

72". 1 


7093 
7177 


7110 

7103 

7275 


7126 
7210 
7292 


7135 

721s 
7300 


7143 

7226 


7152 
7235 
7316 


1 2 
1 2 
1 2 


3 


3 4 

3 4 

3 4 


5 
g 
g 


6 7 8 
6 7 7 
6 6 7 


M 




733° 


7340 




7356 


7364 


7372 


7:: so 




7396 


1 2 


2 


3 4 


5 


6 6 7 



LOGABITHMS. 



185 








1 


2 


3 


4 5 


6 


7 


8 


9 


12 3 


4 5 6 


7 8 9 


55 


7404 


7412 


7419 7427 


7435 7443 


7451 


7459 


7466 


7474 


12 2 


3 4 5 


5 6 7 


56 
57 

58 


7482 
7559 
7634 


7490 
7566 

7642 


7497 
7574 
7649 


7505 

7582 

7657 


7513 7520 
7589 7597 
7664 7672 


7528 
7604 

7679 


7536 
7612 
7686 


7543 
7619 
7694 


7551 

7627 
7701 


12 2 

12 2 
112 


3 4 5 
3 4 5 
3 4 4 


5 6 7 
5 6 7 

5 6 7 


59 
60 
61 


7709 
7782 
7853 


7716 7723 
7789 7796 
7860 '' 7868 


7731 
7803 
7875 


7738 
7810 

7882 


7745 
7.518 
7889 


7752 
7825 
7896 


7760 

7832 
7903 


7767 
7839 
7910 


7774 
7846 
7917 


112 

112 
112 


3 4 4 
3 4 4 

3 4 4 


5 6 7 
5 6 6 

5 6 6 


62 
63 
64 


7924 
7993 
8062 


7931 7938 
8000 8007 

80<39 8075 


7945 
8014 
8082 


7952 
8021 

8089 


7959 
8028 
8096 


7966 
8035 
8102 


7973 
8041 
8109 


7980 
8048 
8116 


7987 
8055 

8122 


112 

112 
112 


3 3 4 
3 3 4 
3 3 4 


5 6 6 
5 5 6 

5 5 6 


65 


8129 


8136 8142 


8149 


8156 


8162 


8169 


8176 


8182 


8189 


112 


3 3 4 5 5 6 


66 
67 
63 


8195 
8261 

8325 


8202 8209 

S267 8274 
8331 1 8338 


8215 
82S0 
8341 


8222 
8287 
8351 


8228 
8293 
8357 


8235 
8299 
8363 


8241 
8306 
8370 


8248 
8312 
8376 


8254 
8319 

8382 


112 
112 
112 


3 3 4 
3 3 4 
3 3 4 


5 5 6 
5 5 6 
4 5 6 


69 
70 
71 


8388 
8451 
8513 


8395 
8457 
8519 


8401 
8463 
8525 


8407 
8470 
8531 


8414 
8476 
8537 


8420 
84S2 
8543 


8426 
8488 

8549 


8432 

8494 
8555 


8439 
8500 
8561 


8415 
8506 
8567 


112 

112 
112 


2 3 4 
2 3 4 
2 3 4 


4 5 6 
4 5 6 
4 5 5 


72 
73 
74 


8573 
8633 
8692 


8579 8585 
8639 S645 
8698 8701 


8591 
8651 
8710 


8597 
8657 
8716 


8603 
8663 

8722 


8609 
8669 

S727 


8615 
8675 
8733 


8621 
8681 
8739 


8627 
S6SG 
8715 


112 
112 
112 


2 3 4 
2 3 4 
2 3 4 


4 5 5 
4 5 5 
4 5 5 


75 


8751 


8756 8762 


876S 


8774 


8779 


S7S5 


8791 


8797 


8802 


112 


2 3 3 


4 5 5 


76 
77 

78 


8808 
8865 
8921 


8814 8820 
8S71 8876 
8927 8932 


8825 
8S82 
8938 


8831 
8887 
8943 


8837 
8893 
8919 


8842 
8899 
8954 


8848 
8904 
8960 


8854 
8910 

8965 


8859 
8915 
8971 


112 

112 

112 


2 3 3 
2 3 3 
2 3 3 


4 5 5 
4 4 5 
4 4 5 


79 
80 
81 


8976 
9031 

9085 


8982 8987 

9036 9042 
9090 9003 


8993 
9047 
9101 


8998 
9053 
9106 


9001 
9058 
9112 


9009 
9063 
9117 


9015 

9069 
9122 


9020 9025 
9074 9079 
9128 9133 


112 

1 1 2 
112 


2 3 3 
2 3 3 
2 3 3 


4 4 5 
4 4 5 
4 4 5 


82 
83 
84 


9138 
9191 
9243 


9143 9149 
9196 9201 
9248 9253 


9154 
9206 
9258 


9159 
9212 
9263 


9165 
9217 

9269 


9170 
9222 

9274 


9175 9180 9186 
9227 9232 923S 
9279 9284 9289 


112 

112 

1 1 2 


2 3 3 
2 3 3 
2 3 3 


4 4 5 
4 4 5 

4 4 5 


85 


9294 


9299 9301 


9309 


9315 


9320 


9325 


9330 9335 9340 


112 


2 3 3 


4 4 5 


86 

87 

88 


9345 
9395 
9445 


9350 
9100 

9450 


9355 
9405 
9455 


9360 
9410 
9460 


9365 
9415 
9465 


9370 
9420 
9469 


9375 
9425 
9474 


9380 9385 
9430 9435 
9479 9484 


9390 
9440 
9489 


112 
Oil 
1 1 


2 3 3 
2 2 3 
2 2 3 


4 4 5 
3 4 4 
3 4 4 


S9 
90 
91 


9494 
9542 

9590 


9499 
9517 
9595 


9501 
9552 
9600 


9509 
9557 

9 '305 


9513 

9562 
9609 


9518 

9566 
9614 


9523 
9571 
9619 


9528 
9576 
9624 


9533 
9581 
9628 


9538 
9586 
9633 


Oil 
Oil 
Oil 


2 2 3 
2 2 3 
2 2 3 


3 4 4 
3 4 4 
3 4 4 


92 
93 
94 


9638 
9685 
9731 


9613 
9689 
9736 


9647 
9694 
9741 


9652 
9699 
9745 


9657 1 9661 
9703 9708 
9750 J 9754 


9666 
9713 
9759 


9671 
9717 
9763 


9675 
9722 
9768 


96S0 
9727 
9773 


Oil 
Oil 
Oil 


2 2 3 
2 2 3 

2 2 3 


3 4 4 
3 4 4 
3 4 4 


95 


9777 


9782 


9786 


9791 


9795 9800 


9805 


9809 9S14 


9818 


Oil 


2 2 3 


3 4 4 


96 
97 

98 


9823 
9868 
9912 


9827 
9S72 
9917 


9832 
9877 
9921 


98.36 
9881 
9926 


9841 ( 9845 
9886 ! 9890 
9930 9934 


9850 
9894 
9939 


9854 9S59 
9899 9903 
9943 9948 


9863 
990S 
9952 


Oil 

Oil 
1 1 


2 2 3 
2 2 3 
2 2 3 


3 4 4 
3 4 4 
3 4 4 


99 


9956 


9961 1 9965 9969 


9974 9978 


9983 


9987 ( 9991 9996 


Oil 


2 2 3 


3 3 4 



180 



NATURAL SINES. 






6' 


12' 


18' 


24' 


30' 


36' 


42' 


48' 


54' 


12 3 


4 5 





0000 


0017 


0035 


0052 


0070 


0087 


0105 


0122 


0140 


0157 


3 6 9 


12 15 


1 

2 

8 


0175 
0349 
0523 


0192 
0966 

0541 


0209 
0384 
0558 


0227 
0401 
0576 


0244 
(HI 9 
0593 


0262 
0436 
0610 


0279 
0454 
0628 


0297 
0471 

0015 


0314 
0488 
0663 


0332 
0506 
0680 


3 6 9 
3 6 9 
3 6 9 


12 15 
12 15 
12 15 


4 0698 

5 0872 
<> 104,") 


0715 
0689 
1063 


0732 

090(5 
1080 


0750 
0924 
1097 


0767 
0941 
1115 


07S5 
0958 
1132 


0802 
0976 
1149 


0819 
0993 
1167 


0837 
1011 

1184 


0854 
1028 
1201 


3 6 9 
3 6 9 
3 6 9 


12 15 
12 14 
12 14 


7 
8 
9 


1219 
1392 

1504 


1236 

1409 

1582 


1253 
1426 
1599 


1271 
1444 
1616 


1288 
1461 
1633 


1305 
1478 
1650 


1323 
1495 
1668 


1340 
1513 
1685 


1357 
1530 
1702 


1374 
1547 
1719 


3 6 9 
3 6 9 
3 6 9 


12 14 
12 14 
12 14 


10 


1736 


1754 


1771 


1788 


1805 


1822 


1840 


1857 


1874 


1891 


3 6 9 


12 14 


11 

18 
18 


1908 
2079 
2250 


1925 
2096 
2267 


1942 
2113 

2284 


1959 
2130 

2300 


1977 
2147 
2317 


1994 
2164 
2334 


2011 
2181 
2351 


2028 
2198 
2368 


2045 
2215 
2385 


2062 
2232 
2402 


3 6 9 
3 6 9 
3 6 8 


11 14 
11 14 
11 14 


14 

15 ! 
16 


2419 
2588 
2756 


2436 
2605 
2773 


2453 
2622 
2790 


2470 
2639 
2807 


2487 
2656 
2823 


2504 
2672 
2840 


2521 
2689 

2S57 


2538 
2706 

2874 


2554 
2723 
2890 


2571 
2740 
2907 


3 6 8 
3 6 8 
3 6 8 


11 14 
11 14 
11 14 


17 

18 
10 


2921 

3090 
3256 


2940 
3107 
3272 


2957 
3123 

3289 


2974 
3140 
3305 


2990 
3156 
3322 


3007 
3173 

3338 


3024 
3190 
3355 


3040 
3206 
3371 


3057 
3223 

3387 


3074 
3239 
3404 


3 6 8 
3 6 8 
3 5 8 


11 14 
11 14 
11 14 


20 


3420 


3437 


3453 


3469 


3486 


3502 


3518 


3535 


3551 


3567 


3 5 8 


11 14 


21 
22 

23 


3584 
3746 
3907 


3600 
3762 
3923 


3616 

3778 
3939 


3633 
3795 
3955 


3649 
3811 
3971 


3665 
3827 
3987 


3681 
3843 
4003 


3697 
3859 
4019 


3714 
3875 
4035 


3730 
3891 
4051 


3 5 8 
3 5 8 
3 5 8 


11 14 
11 14 
11 14 


24 
25 
26 


4067 
4226 
4384 


4083 
4242 
4399 


4099 

4258 
4415 


4115 
4274 
4431 


4131 

4289 
4446 


4147 
4305 
4462 


4163 
4321 

4478 


4179 
4337 
4493 


4195 
4352 
4509 


4210 

4368 
4524 


3 5 8 
3 5 8 
3 5 8 


11 13 
11 13 
10 13 


27 

28 
29 


4540 
4695 
4848 


4555 
4710 
4863 


4571 
4726 

4879 


4586 
4741 
4894 


4602 
4756 
4909 


4617 
4772 
4924 


4633 

4787 
4939 


4648 
4802 
4955 


4664 
4818 
4970 


4679 
4833 

4985 


3 5 8 
3 5 8 
3 5 8 


10 13 
10 13 
10 13 


30 


5000 


5015 


5030 


5045 


5060 


5075 


5090 


5105 


5120 


5135 


3 5 8 


10 13 


31 
32 
88 


5150 
5299 
5446 


5165 
5314 
5461 


5180 
5329 
5476 


5195 
5344 
5490 


5210 
5358 

5505 


5225 
5373 
5519 


5240 
5388 
5534 


5255 
5402 
5548 


5270 
5417 
5563 


5284 
5432 
5577 


2 5 7 
2 5 7 
2 5 7 


10 12 

10 12 
10 12 


84 

35 
36 ! 


5736 

5878 


5606 
5750 
5892 


5621 
5764 

5901 ; 


5635 

5779 
5920 


5650 
5793 
5934 


5664 
5807 
594S 


5678 
5821 
5962 


5693 
5835 
5976 


5707 
5850 
5990 


5721 
5864 
6004 


2 5 7 
2 5 7 
2 5 7 


10 12 
10 12 
9 12 


157 
88 
89 


6018 

6157 


60321 6046 
6170 6184 
6307 6320 


6000 
6198 


6074 
6211 
6347 


6088 
6225 
6361 


6101 
6239 
6374 


6115 
6252 
6388 


6129 
6266 
6401 


6143 
6280 
6414 


2 5 7 
2 5 7 
2 4 7 


9 12 
9 11 
9 11 


10 6428 


6441 6455 


6468 


G481 


6494 


6506 


6521 


6534 




2 4 7 


9 11 


11 
19 
18 

44 




6574 6587 
6704 0717 


6730 


6613 
6743 


6626 
6756 


6639 
6769 

6896 


6652 
67S2 
6909 


6665 

6794 
6921 


6678 
6S07 
6934 


2 4 7 
2 4 6 
2 4 6 


9 11 
9 11 
8 11 




6972 




G997 


7009 


7022 


7034 


7046 


7059 


2 4 6 


8 10 



_v./;. — N umbers In difference-columns to be subtracted, not added. 



NATURAL SIXES. 



187 



0' 


6' 12' i 18' 


24' 


3(y 


36' 


42' 


48' 


54' 


1 


2 


3 


4 


5 


45 c 


7071 


7083 7096 7108 


7120 


7133 


7145 


7157 


7169 


7181 


2 


4 


6 


8 


10 


46 


7193 


7206 7218 


7230 


7242 


7254 


7266 


7278 


7290 


7302 


2 


4 


6 


8 


10 


47 


7314 


7325 7337 


7349 


7361 


7373 


7385 


7396 


7408 


7420 


o 


4 


6 


8 


10 


48 


7431 


7443 7455 


7466 


7478 


7490 


7501 


7513 


7524 


7536 


2 


4 


6 


8 


10 


49 


7547 


7558 7570 7581 


7593 


7604 


7615 


7627 


7638 


7649 


2 


4 


6 


8 


9 


50 


7660 


7672 7683 


7694 


7705 


7716 


7727 


7738 


7749 


7760 


2 


4 


6 


7 


9 


51 


7771 


7782 7793 


7804 


7815 


7826 


7837 


7848 


7859 


7869 


2 


4 


5 


7 


9 


52 


7880 


7S91 7902 


7912 


7923 


7934 


7944 


7955 


7965 


7976 


2 


4 


5 


7 


9 


53 


7986 


7997| 8007 S018 


8028 


8039 


8049 


8059 


8070 


8080 


2 


3 


5 


7 


9 


54 


809U 


8100 8111 1 8121 


8131 


8141 


8151 


8161 


8171 


8181 


2 


3 


5 


7 


8 


55 


8192 


S202 


8211 


8221 


8231 


8241 


8251 


8261 


8271 


8281 


2 


3 


5 


7 


8 


56 


8290 


8300 


8310 


8320 


8329 


8339 


8348 


8358 


836S 


8377 


2 


3 


5 


6 


8 


57 


8387 


8396 S406 


8415 


8425 


8434 


8443 


8453 


8462 


8471 


o 


3 


5 


6 


8 


58 


8480 


8490 


8499 8508 


8517 


8526 


S536 


8545 


8554 


8563 


2 


3 


5 


6 


8 


59 


8572 


.8581 


8590 


8599 


8607 


8616 


8625 


8634 


8643 


8652 


1 


3 


4 


6 


7 


60 


8660 


8669 


8678 


8686 


8695 


8704 


8712 


S721 


8729 


8738 


1 


3 


4 


6 


7 


61 


8746 


8755 


S763 


8771 


8780 


8788 


8796 


8805 


8813 


8821 


1 


3 


4 


6 


7 


62 


8829 


8838 


8846 


8854 


8862 


8870 


8878 


8SS6 


8894 


8902 


1 


3 


4 


5 


7 


63 


8910 


8918 


8926 


8934 


8942 


8949 


8957 


8965 


8973 


8980 


1 


3 


4 


5 


6 


64 


898S 


8996 9003 


9011 


9018 


9026 


9033 


9041 


9048 


9056 


1 


3 


4 


5 


6 


65 


9063 


9070 


907S 


90-85 


9092 


9100 


9107 


9114 


9121 


9128 


1 


2 


4 


5 


6 


66 


9135 


9143 


9150 


9157 


9164 


9171 


9178 


9184 


9191 


9198 


1 


2 


3 


5 


6 


67 


9203 


9212 


9219 


9225 


9232 


9239 


9245 


9252 


9259 


9265 


1 


2 


3 


4 


6 


68 


9272 


9278 


9285 


9291 


9298 


9304 


9311 


9317 


9323 


9330 


1 


o 


3 


4 


5 


69 


9336 


9342 


934S 


9354 


9361 


9367 


9373 


9379 


9385 


9391 


1 


2 


3 


4 


5 


70 


9397 


9403 


94C9 


9415 


9421 


9426 


9432 


9438 


9444 


9449 


1 


2 


3 


4 


5 


71 


9455 


9461 


9466 


9472 


9478 


9483 


9489 


9494 


9500 


9505 


1 


2 


3 


4 


5 


72 


9511 


9516 


9521 


9527 


9532 


9537 


9542 


9548 


9553 


9558 


1 


2 


3 


4 


4 


73 


9563 


9568 


9573 


9578 


9583 


9588 


9593 


9598 


9603 


9608 


1 


2 


2 


3 


4 


74 


9613 


9617 


9622 9627 


9632 


9636 


9641 


9646 


9650 


9655 


1 


2 


2 


3 


4 


75 


9659 


9664 


966S 


9673 


9677 


9681 


9686 


9690 


9694 


9699 


1 


1 


2 


3 


4 


76 


9703 


9707 


9711 


9715 


9720 


9724 


9728 


9732 


9736 


9740 


1 


1 


2 


3 


3 


77 


9744 


9748 9751 


9755 


9759 


9763 


9767 


9770 


9774 


9778 


1 


1 


2 


3 


3 


7S 


9781 


9785 


9789 


9792 


9796 


9799 


9803 


9S06 


9810 


9813 


1 


1 


2 


2 


3 


79 


9S1G 


9S20 


9823 


9S26 


9829 


9S33 


9836 


9S39 


9842 


9845 


1 


1 


2 


2 


3 


80 


9848 


9851 


9854 


9S57 


9860 


9863 


9866 


9869 


9871 


9874 





1 




2 


2 


81 


9877 


9SS0 


9882 


9SS5 


9888 


9890 


9893 


9895 


9898 


9900 





1 




2 


2 


82 


9903 


9905 


9907 9910 


9912 


9914 


9917 


9919 


9921 


9923 





1 




2 


2 


83 


9925 


9928 9930 9932 


9934 


9936 


9938 


9940 


9942 


9943 





1 




1 


2 


84 


9945 


9947 


9949 9951 


9952 


9954 


9956 


9957 


9959 


9960 





1 




1 


1 


85 


9962 


9963 


9965 9966 


9968 


9969 


9971 


9972 


9973 


9974 










1 


1 


86 


9976 


9977 


9978 9979 


9980 


9981 


9982 


9983 


9984 


9985 










1 


1 


S7 


99S6 


9987 9988 99S9 


9990 


9990 


9991 


9992 


9993 


9993 











1 


1 


88 


9994 


9995 9995 9996 


9996 


9997 


9997 


9997 


9998 


9998 

















89 


999S 


9999 9099 9999 
1 1 


9999 


1.000 
nearly 


1.000 
nearly 


1.000 
nearly 


1.000 
nearly 


1.000 
nearly 


















188 



NATURAL COSINES. 





/ 


G' 


12' 


IS' 


24' 


30' 


36' 


42' 


48' 


54' 


12 3 


4 5 





1-000 


1-000 
oearly. 


1-000 
nearly. 


t-ooo 

oearly. 


1-000 
oearly. 


9999 


9999 


9999 


9999 


9999 








1 

.J 
8 


9998 

9994 


9993 


99S4 


9997 
9992 


9997 
9991 
9982 


9997 
9990 
9981 


9996 
9990 
9980 


9996 

99S!) 
9979 


9995 
99SS 
9978 


9995 
9987 
9977 




1 




1 1 
1 1 


4 
5 

6 


9976 
9962 

9945 


9974 
9960 
9943 


9973 

9959 
9942 


9972 

9957 
9940 


9971 

9956 
9938 


9969 
9954 
9936 


9968 
9952 
9934 


9966 
9951 
9932 


9965 
9949 
9930 


9963 
9947 

992S 


1 
1 1 
1 1 


1 1 
1 2 
1 2 


7 
8 
9 


9925 

9903 

9877 


9923 
9900 
9874 


9921 
9898 
9871 


9919 

9895 
9869 


9917 
9S93 

9866 


9914 
9890 
9863 


9912 

9888 
9860 


9910 

9S85 
9857 


9907 
9882 
9854 


9905 
9880 
9851 


1 1 
1 1 
1 1 


2 2 
2 2 

2 2 


10 


9848 


9845 


9842 


9S39 


9S36 


9833 


9829 


9826 


9823 


9820 


1 1 2 


2 3 


11 
12 
13 


981(3 
9781 
9744 


9813 
9778 
9740 


9810 
9774 
9736 


9806 
9770 
9732 


9803 
9767 
9728 


9799 
9763 
9724 


9796 
9759 

9720 


9792 
9755 
9715 


9789 
9751 
9711 


9785 
9748 
9707 


1 1 2 
112 
1 1 2 


2 3 

3 3 
3 3 


14 
15 
16 


9703 
9659 
9613 


9699 
9055 
9608 


9694 
9650 
9603 


9690 
9646 
9598 


9686 
9641 
9593 


9681 
9636 

9588 


9677 
9632 
9583 


9673 
9627 

9578 


9068 
9622 
9573 


9664 
9617 
9568 


1 1 2 
12 2 
1 2 2 


3 4 
3 4 
3 4 


17 
18 
19 


9563 
9511 
9455 


9558 
9505 
9449 


9553 
9500 
9444 


9548 
9494 
9438 


9542 
9489 
9432 


9537 
9483 
9426 


9532 
9478 
9421 


9527 
9472 
9415 


9521 
9466 
9409 


9516 
9461 
9403 


12 3 
12 3 
12 3 


4 4 
4 5 
4 5 


20 


9397 


9391 


9385 


9379 


9373 


9367 


9361 


9354 


9348 


9342 


12 3 


4 5 


21 
22 
28 


9336 
9272 
9205 


9330 
9265 
9198 


9323 
9259 
9191 


9317 
9252 
9184 


9311 
9245 
9178 


9304 
9239 
9171 


9298 
9232 
9164 


9291 

9225 
9157 


9285 
9219 
9150 


9278 
9212 
9143 


12 3 
12 3 
12 3 


4 5 

4 6 

5 6 


24 
25 
26 


9135 
9063 

SD.SS 


9128 
9056 
8980 


9121 
9048 
8973 


9114 
9041 
8965 


9107 
9033 

8957 


9100 
9026 
8949 


9092 
9018 
8942 


9085 
9011 
8934 


9078 
9003 

8926 


9070 
8996 
8918 


12 4 

13 4 

1 3 4 


5 6 
5 6 
5 6 


27 

28 
29 


8910 
8829 
8740 


8902 

8821 
8738 


8894 
8813 
8729 


8886 
8805 
8721 


8878 
8796 
8712 


8870 
8788 
8704 


8862 
8780 
8095 


8854 
8771 
8686 


8846 
8763 
8678 


8838 
8755 
8669 


1 3 4 
1 3 4 
1 3 4 


5 7 

6 7 

6 7 


80 


8660 


8662 


8643 


8634 


8625 


8616 


8607 


8599 


8590 


8581 


1 3 4 


6 7 


81 

82 


8572 
8480 

s:;s7 


8563 
8471 

8377 


8554 
8462 
8368 


8545 
8453 
8358 


8536 
8443 

8348 


8526 
8434 
8339 


8517 
8425 
8329 


8508 
8415 
8320 


8499 
8406 
8310 


8490 
8396 
8300 


2 3 5 
2 3 5 
2 3 5 


6 8 
6 8 
6 8 


84 

86 

86 


8290 
8192 

•sooo 


8281 
8181 
8080 


8271 
8171 
8070 


8261 
8161 


8251 
8151 

8049 


8241 
8141 

8039 


8231 
8131 
8028 


8221 
8121 
8018 


8211 
8111 
8007 


8202 
8100 
7997 


2 3 5 
2 3 5 
2 3 5 


7 8 
7 8 
7 9 


37 

39 


7880 
7771 


7976 
7760 


7965 
7859 

7749 


7956 
7848 

7738 


7944 
7837 
7727 


7934 
7826 
7716 


7923 

7815 
7705 


7912 
7804 
7694 


7902 
7793 
7683 


7891 

7782 
7672 


2 4 5 
2 4 5 
2 4 6 


7 9 
7 9 
7 9 


40 


7660 


7649 


7638 


7627 


7615 


7604 


7593 


7581 


7570 


7559 


2 4 6 


8 9 


41 
42 

415 


7314 


7302 


7524 
7290 


7513 
7278 


7501 
7266 


7490 
7373 


7478 
7361 
7242 


7466 
7349 

7230 


7455 
7337 
7218 


7443 
7.'525 
7206 


2 4 6 
2 4 6 
2 4 6 


8 10 
8 10 
8 10 


it 


7193 


7181 




7157 


7115 




7120 


7108 


7096 


7083 


2 4 6 


8 10 



X.P>. — Numbers in diii'ereiice-coluinns to be subtracted, not added. 



NATURAL COSIXES. 



189 





0' 


6 / 


12' 


18' 


24/ 


3(y 


36' 


42' 48' 


54' 


12 3 


4 5 


45° 


7071 


7059 


7046 


7034 


7022 


7009 


6997 


6984 


6972 


6959 


2 4 6 


8 10 


46 
47 

48 


0947 
6S20 
6691 


6934 

6S07 
6678 


6921 
6794 
6665 


6909 

6782 
6652 


6SdG 
6769 
6639 


68S4 
6756 
6626 


6871 
6743 
6613 


6858 
6730 
6600 


6845 
6717 

65S7 


6833 
6704 
6574 


2 4 6 
2 4 6 
2 4 7 


8 11 

9 11 
9 11 


49 
50 
51 


6561 
6428 

6293 


6547 
6414 
6280 


6534 
6401 
6266 


6521 
638S 

6252 


6508 
6374 

6239 


6494 
6361 
6225 


6481 
6347 
6211 


646S 
6334 
6198 


6455 
6320 
61S4 


6441 
6307 
6170 


2 4 7 
2 4 7 
2 5 7 


9 11 
9 11 
9 11 


52 
53 
54 


6157 
6018 
5878 


6143 
6004 

5864 


6129 
5990 
5850 


6115 
5976 
5835 


6101 
5962 
5821 


6088 
5948 
5807 


6074 
5934 
5793 


6060 
5920 
5779 


6046 
5906 
5764 


6032 
5S92 
5750 


2 5 7 
2 5 7 
2 5 7 


9 12 
9 12 
S i2 


55 


5736 


5721 


5707 


5693 


5678 


5664 


5650 


5635 


5621 


5606 


2 5 7 


10 12 


56 
57 

58 


5592 
5446 

5299 


5577 
5432 
5284 


5563 
5417 
5270 


554S 
5402 
5255 


5534 

5388 
5240 


5519 
5373 

5225 


5505 
5358 
5210 


5490 
5344 

5195 


5476 
5329 
5180 


5461 
5314 
5165 


2 5 7 
2 5 7 
2 5 7 


10 12 
10 12 
10 12 


59 
60 
61 


5150 
5000 

4848 


5135 
4985 
4833 


5120 
4970 
4818 


5105 
4955 
4802 


5090 
4939 

4787 


5075 
4924 
4772 


5060 

4909 
4756 


5045 
4S94 
4741 


5030 

4879 
4726 


5015 

4863 
4710 


3 5 8 
3 5 8 

3 5 8 


10 13 
10 13 
10 13 


62 

63 
64 


4695 
4540 
4384 


4679 

4524 
4368 


4664 
4509 
4352 


464S 
4493 
4337 


4633 
447S 
4321 


4617 

4462 
4305 


4602 
4446 
4289 


4586 
4431 

4274 


4571 4555 
4415 4399 
4258 4242 


3 5 8 
3 5 8 
3 5 8 


10 13 

10 13 

11 13 


65 


4226 


4210 


4195 


4179 


4163 


4147 


4131 


4115 4099 4083 


3 5 8 


11 13 


66 
67 
68 


4067 
3907 
3746 


4051 
3891 
3730 


4035 
3875 
3714 


4019 
3S59 
3697 


4003 
3843 
3681 


3987 
3827 
3665 


3971 
3811 
3649 


3955 3939 3923 
3795 3778 3762 
3633 3616 3600 


3 5 8 
3 5 8 
3 5 8 


11 14 

11 14 
11 14 


69 
70 
71 


35S4 
3420 
3256 


3567 
3404 
3239 


3551 
3387 

3223 


3535 
3371 
3206 


3518 
3355 
3190 


3502 
333S 
3173 


3486 
3322 

3156 


3469 3453 3437 
3305 3289 3272 
3140 3123 3107 


3 5 8 
3 5 8 
3 6 8 


11 14 
11 14 
11 14 


72 
73 
74 


3090 
2924 
2756 


3074 
2907 
2740 


3057 
2890 
2723 


3040 

2S74 
2706 


3024 
2857 
2689 


3007 

2840 
2672 


2990 
2S23 
2656 


2974 2957 
2807 2790 
2639 2622 


2940 
2773 
2605 


3 6 8 
3 6 8 
3 6 8 


11 14 
11 14 
11 14 


75 


25S8 


2571 


2554 


2538 


2521 


2504 


2487 


2470 2453 


2436 


3 6 8 


11 14 


76 
77 

78 


2419 

2250 
2079 


2402 
2233 
2062 


2385 
2215 
2045 


2368 
2198 
2028 


2351 2334 
21S1 2164 
2011 1994 


2317 
2147 
1977 


2300 2284 
2130 2113 
1959J 1942 


2267 
2096 
1925 


3 6 S 
3 6 9 
3 6 9 


11 14 

11 14 
11 14 


79 
80 

81 

S2 
83 

S4 


1908 
1736 
1564 


1891 
1719 
1547 


1874 
1702 
1530 


1S57 
1685 
1513 


1840 1822 
1668 1650 
1495 1478 


1805 
1633 
1461 


1788 1771 1754 
1616 1599, 15S2 
1444 1426 1 1409 


3 6 9 
3 6 9 
3 6 9 


12 14 
12 14 
12 14 


1392 
1219 
1045 


1374 
1201 
1028 


1357 
1184 
1011 


1340 

1167 
0993 


1323 1305 
1149 1132 
0976 | 0958 


1288 
1115 
0941 


1271 1253 1236 
1097 1080 1063 
0924 0906 08S9 


3 6 9 
3 6 9 
3 6 9 


12 14 
12 14 
12 14 


So 


0S72 


0S54 


0837 


0S19 


0802 | 0785 


0767 


0750 0732 0715 


3 6 9 


12 15 


86 

S7 
88 


069S 
0523 
0349 


0680 
0506 
0332 


0663 
0488 
0314 


0645 
0471 

0297 


0628 0610 
0454 0436 
0279 0262 


0593 

0419 
0244 


0576 0558 0541 
0401 0384 0366 
0227 0209 0192 


3 6 9 
3 6 9 
3 6 9 


12 15 
12 15 
12 15 


89 


0175 


0157 


0140 


0122 


0105 00S7 0070 


0052 0035 001 7 ' 3 6 9 


12 15 



X.B. — Numbers in difference-columns to be subtracted, not added. 



190 



.Y. 1 TUB* I L 1\ I X GENTS. 





0' 


6' 


12' | 


18' 


24' 


30' 


36' 


42' 


48' 


54' 


1 


2 3 


4 


5 


0° 


•0000 


0017 


0035 


0052 


0070 


0087 


0105 


0122 


0140 


0157 


3 


6 9 


12 


14 


1 


•oi::> 


0192 


0209 


0227 


0244 


02G2 


0279 


0297 


0314 


0332 


3 


6 9 


12 


15 


o 


•0349 


0367 


0384 


0102 


0419 


01:57 


0454 


0472 


0489 


0507 


3 


6 9 


12 


15 


» 


•0624 


0542 


0659 


0577 


0594 


0G12 


0G29 


0647 


0GG4 


0082 


3 


6 9 


12 


15 


4 


•0699 


0717 


0734 


0752 


0769 


0787 


0805 


0822 


0840 


0857 


3 


6 9 


12 


16 


5 


■0875 


0892 


0910 


0928 


0945 


09G3 


0981 


0998 


1016 


1033 


3 


6 9 


12 


15 




•1051 


10G9 


lose 


1104 


1122 


1139 


1157 


1175 


1192 


1210 


3 


6 9 


12 


15 


7 


•1228 


124G 


1263 


1281 


1299 


1317 


1334 


1352 


1370 


1388 


3 


6 9 


12 


15 


8 


•1405 


1423 


1441 


1459 


1477 


1495 


1512 


1530 


1548 


156G 


3 


6 9 


12 


15 


9 


1584 


1002 


1G20 


1638 


1655 


1G73 


1G91 


1709 


1727 


1745 


3 


6 9 


12 


15 


10 


•17G3 


1781 


1799 


1817 


1835 


1853 


1871 


1890 


1908 


1926 


3 


6 9 


12 


15 


11 


•1944 


19G2 


1980 


1998 


2016 


2035 


2053 


2071 


2089 


2107 


3 


6 9 


12 


15 


12 


•2126 


2144 


21G2 


21S0 


2199 


2217 


2235 


2254 


2272 


2290 


3 


6 9 


12 


15 


13 


•2309 


2327 


2345 


23G4 


2382 


2401 


2419 


2438 


245G 


2475 


3 


6 9 


12 


15 


14 


•2493 


2512 


2530 


2549 


2568 


2586 


2G05 


2623 


2642 


26G1 


3 


6 9 


12 


16 


15 


•2079 


2G9S 


2717 


2736 


2754 


2773 


2792 


2811 


2830 


2849 


3 


6 9 


13 


16 


16 


•2867 


2886 


2905 


2924 


2943 


29G2 


2981 


3000 


3019 


3038 


3 


6 9 


13 


16 


17 


•3057 


3076 


309G 


3115 


3134 


3153 


3172 


3191 


3211 


3230 


3 


6 10 


13 


10 


18 


•3249 


32G9 


3288 


3307 


3327 


3346 


33G5 


3385 


3404 


3424 


3 


6 10 


13 


16 


19 


•3443 


34G3 


3482 


3502 


3522 


3541 


35G1 


3581 


3600 


3620 


3 


6 10 


13 


17 


'JO 


•3G40 


3059 


3G79 


3G99 


3719 


3739 


3759 


3779 


3799 


3819 


3 


7 10 


13 


17 


21 


•3839 


3859 


3879 


3899 


3919 


3939 


3959 


3979 


4000 


4020 


3 


7 10 


13 


17 


22 


•4040 


4061 


4081 


4101 


4122 


4142 


41G3 


4183 


4204 


4224 


3 


7 10 


14 


17 


23 


•4245 


42G5 


4286 


4307 


4327 


4348 


4369 


4390 


4411 


4431 


3 


7 10 


14 


17 


24 


•4452 


4473 


4494 


4515 


4536 


4557 


4578 


4599 


4621 


4642 


4 


7 10 


14 


18 


25 


•4GG3 


4GS4 


4706 


4727 


4748 


4770 


4791 


4813 


4834 


4856 


4 


7 11 


14 


18 


26 


•4877 


4899 


4921 


4942 


4964 


4986 


5008 


5029 


5051 


5073 


4 


7 11 


15 


18 


27 


•5095 


5117 


5139 


5161 


5184 


5206 


5228 


5250 


5272 


5295 


4 


7 11 


15 


18 


28 


•5317 


5340 


53G2 


5384 


5407 


5430 


5452 


5475 


5498 


5520 


4 


8 11 


15 


19 


29 


•5643 


55GG 


5589 


5612 


5635 


5658 


5G81 


5704 


5727 


5750 


4 


8 12 


15 


19 


30 




5797 


5820 


5844 


58G7 


5890 


5914 


5938 


59G1 


5985 


4 


8 12 


16 


20 


31 


•6000 


6032 


G056 


6080 


G104 


G128 


G152 


617G 


G200 


6224 


4 


8 12 


16 


20 


89 


•6249 


6273 


6297 


6322 


6346 


0371 


G395 


6420 


G445 


6469 


4 


8 12 


16 


20 


88 

34 


-6494 


6519 


6544 


65G9 


6594 


6G19 


6G44 


6G69 


6694 


6720 


4 


8 13 


17 


21 




G771 


6796 


G822 


6847 


6873 


G899 


6924 


6950 


6976 


4 


9 13 


17 


21 


8fi 


•7002 


7028 


7054 


7080 


7107 


7133 


7159 


718G 


7212 


7239 


4 


9 13 


18 


22 


36 




7292 


7319 


7346 


7373 


7400 


7427 


7454 


7481 


7508 


5 


9 14 


18 


23 


37 


•753G 




7590 


7618 


7G46 


7673 


7701 


772!) 


7757 


7785 


5 


9 14 


18 


23 


88 


•7813 


7841 


7869 


7898 


792G 


7954 


7983 


8012 


8O40 


8069 


5 10 14 


19 


24 


89 


•8098 


8127 


8156 


8185 


8214 


8243 


8273 


8302 


8332 


8361 


5 


10 15 


20 


24 


40 


•8391 


8421 


8451 




8511 


8541 


S571 


8G01 


8632 


8662 


5 


10 15 


20 


25 


41 






8754 




8816 


8847 


8878 


891C 


8941 


8972 


5 


10 16 


21 


26 


ia 


•9004 


9036 


9007 


9099 


9131 


9163 


919: 


9228 


9260 


929: 


5 


11 16 


21 


27 


48 


•9326 






9424 




9490 


9522 


!)",( 


959( 


9622 


6 11 17 


22 


28 


41 


1 -9G57 


9091 


9725 


9759 


9793 


| 9827 


9861 


9894 


99: K 


9965 


6 11 17 


23 


29 



NATURAL TANGENTS. 



191 



<y 


6' 


12' 


18' 


24' 


30' 


36' 


42' 


48' 


54' 


12 3 4 5 


45° 


1-0000 


0035 


0070 


0105 


0141 


0176 0212 


0247 


0283 


0319 


6 12 IS 


24 30 


46 
48 


1-0355 

1-0724 

' 1-1106 


0392 
0761 
1145 


0428 
0799 
1184 


0464 
0837 
1224 


0501 0538 
0875 0913 
1263 1303 


0575 
0951 
1343 


0612, 0649 
0990 1028 

1383 1423 


06S6 
1067 
1463 


6 12' 18 

6 13 19 

7 13 20 


25 31 

25 32 

26 33 


49 1-1504 

50 1-1918 

51 1-2349 


1544 
1960 
2393 


15S5 
2002 
2437 


1626 
2045 
2482 


1667 
2088 
2527 


1708 
2131 

2572 


1750 
2174 
2617 


1792 
2218 
2662 


1833 
2261 
2708 


1875 
2305 
2753 


7 14 21 28 34 

7 14 22 29 36 

8 15 23 30 38 


52 1-2799 

53 !' 1-3270 

54 ' 1-3764 


28461 2S92 

3319 3367 
3814 3865 


2938 
3416 
3916 


2985 1 3032 
3465 3514 
3968 4019 


3079 
3564 
4071 


3127 
3613 
4124 


3175 
3663 
4176 


3222 
3713 
4229 


8 16 23 

8 16 25 

9 17" 26 


31 39 

33 41 

34 43 


55 1-4281 


4335' 4388 


4442 


4496 4550 


4605 


4659 


4715 


4770 


9 18 27 36 45 


56 1-4826 

57 1-5399 

58 1-6003 


4882 4935 
5458 ool, 
6066 6128 


4994 
5577 
6191 


5051 
5637 
6255 


5108 
5697 
6319 


5166 
5757 
6383 


5224 
5818 
6447 


5232 5340 
5880 5941 
6512 6577 


10 19 29 

10 20 30 

11 21 32 


38 48 
40 50 
43 53 


59 
60 
61 


1-6643 
1-7321 

1-8040 


6709 6775 6842 
7391 7461 7532 
8115 8190 8265 


6909 
7603 
S341 


6977 
7675 

8418 


7045 
7747 
8495 


7113 1 7182 7251 
7820 7893 7966 
8572 ' 8650 1 8728 


11 23 34 

12 24 36 

13 26 38 


45 56 
48 60 
51 64 


62 
68 
64 


1-8807 
1-9626 
2-0503 


8887 

9711 

0594 


8967 
9797 

0686 


9047 
9833 

0778 


9128 
9970 
0872 


9210 

0057 
0965 


9292 
0145 
1060 


9375 
0233 
1155 


9458 9542 
0323 ' 0413 
1251 1348 


14 27 41 

15 29 44 

16 31 47 


55 68 
58 73 
63 78 


65 


2-1445 


1543 1642 1742 


1842 


1943 


2045 


2148 


2251 2355 


17 34 51 


68 85 


66 

67 1 
6S 


2-2460 
2-3559 
2-4751 


2566 
3673 
4S76 


2673 2781 
3789 3906 
5002 1 5129 


2889 
4023 
5257 


2998 
4142 
5386 


3109 
4262 
5517 


3220 
4383 
5649 


3332 3445 
4504 4627 
5782 5916 


18 37 55 

20 40 60 
22 43 65 


74 92 

79 99 
87 108 


69 
70 

71 


, 2-6051 
2-7475 
2-9042 


6187 
7625 
9208 


6325 6464 
7776 7929 
9375| 9544 


6G05 67461 6SS9 
8083 8239 8397 

9714 9887 0061 


7034! 7179 
8556| 8716 
0237 0415 


7326 
8578 
0595 


24 47 71 
26 52 73 
29 58 87 


95 118 
104 130 
115 144 


72 
73 

74 


1 3-0777 

3-2709 

; 3-4S74 


0961 
2914 
5105 


1146 

3122 
5339 


1334 
3332 
5576 


152l! 1716 1910 
3544 1 3759 3977 
5816| 6059 6305 


2106 
4197 

6554 


2305 
4420 
6806 


2506 

4646 
7062 


32 64 96 

36 72 108 

41 82 122 


129 161 
144 180 
162 203 


75 


3-7321 


7583 7848 


8118 


8391 


8667 


8947 


9232 


9520 


9812 


46 94 139 


186 232 


76 
77 

78 


' 4-0108 

4-3315 

t 4-7046 


0403 0713 
3662 4015 
7453 7867 


1022 
4374 
S288 


1335 
4737 
8716 


1653 
5107 
9152 


1976 
548? 
9594 


2303 
5864 
0045 


2635 
6252 
0504 


2972 
6646 
0970 


53 107 160 
62 124 186 
73 146 219 


214 267 
248 310 
292 365 


79 

80 

SI 


5-1446 
5-6713 
6-3138 


1929 
7297 
3859 


2422 
7894 
4596 


2924 
8502 
5350 


3435 
9124 
6122 


3955 


44S6 


5026 


5578 


6140 
2432 
0264 


87 175 262 350 437 


9758 0405 
6912 7920 


1066 1.42 
8548 9395 


Difference-columns 
cease to be useful, 
owing to the rapid- 
ity with which the 
value of the tangent 
changes. 


S2 
83 
84 


7-1154 
S-1443 
9-5144 


2066 
2*336 
9-677 


3002 3962 
38631 5126 

9-845 10-02 


4947 5958 6996 
6427 7769 9152 
10-20 10-39 10-58 


8062 9158 
0579 2052 
10-78 10-99 


0285 
3572 
11-20 


85 


11-43 


11-66 ll-9l!l2-16 


12-43 12-71 13-00 


13-30 13-62 


13-95 

18-46 
27*27 

52-08 


86 
87 
88 


14-30 

19-08 

1 28-64 


14-67 15*06 15'46 
19-74 20-45 21-20 
30-14 31-82 33-69 


15-89 16-35 16-83 
22-02 22 90 23-86 
35-80 38-19 40-92 


17-3417-89 

24-90 26-03 
44-07 47.74 


89 57-29 


63-66 71-62 81 85 


95-49 114-6 143-2 

1 


191-0 286-5 573-0 







19: 



y. 1 TUBAL COT. IXGKXTS. 





(X 


6 / 


12' 


18' 


24' 


30" 


36' 


42' 
81-86 


48' 


54' 
63-66 






0° 


j Inf. 




286*6 


1910 


143 2 


114-0 


95-49 


71-62 


Difference-columns 
not useful here, ow- 
ing to the rapidity 
with which the val- 
ue of the cotangent 
changes. 


g 
8 


67*29 
28*64 
19-08 


52-08 
2< "2» 
18-4(5 


47 74 
26-03 

17-89 


44 07 
24-90 
17-34 


40-92 
23-86 
16-83 


38-19 

22-90 
16-35 


35-80 
22-02 
15-89 


33-69 
21-20 
15-46 


31-82 
20-46 

15-06 


30-14 
19-74 
1467 


4 
5 
6 


14-30 

1143 

9-5144 


13-95 
11-20 

3572 


13 62 
1099 

2052 


13-30 
10-78 

0579 


13 00 
10-58 
9152 


12-71 
10 39 
7769 


12-43 

10 20 
6427 


12-16 
10-02 
5126 


11-91 

9-S45 
3863 


11-66 
9-677 
2636 


7 

8 
<) 


8-1443 
7*1154 

£•9198 


0285 
0264 
2432 


9158 
9395 
1742 


8062 
8548 
1066 


6996 

7920 
0405 


5958 
6912 
9758 


4947 
6122 
9124 


3962 
5350 
8502 


3002 
4596 

7894 


2066 
3859 
7297 




12 3 


4 5 


10 


5*6713 


6140 


5578 


5026 


448G 


3955 


3435 


2924 


2422 


1929 


11 
18 
18 


5-1446 
4-7046 

4-3315 


0970 
6046 
2972 


0504 
6252 
2635 


0045 
5864 
2303 


9594 
5483 
1976 


9152 
5107 

1653 


8716 
4737 
1335 


8288 
4374 
1022 


7867 
4015 
0713 


7453 
3662 
0408 


74 148 222 
63 125 188 
53 107 160 


296 370 
252 314 
214 267 


14 
15 

16 


4-0108 
3-7321 
3-4874 


9812 
7062 
4646 


9520 

6S06 
4420 


9232 
6554 
4197 


8947 
6305 
3977 


8667 
6059 
3759 


8391 
5816 
3544 


8118 
5576 
3332 


7848 
5339 
3122 


7583 
5105 
2914 


46 93 139 
41 82 122 
36 72 108 


186 232 
163 204 
144 180 


17 

18 
19 


3-2709 
30777 
2-0042 


250G 
0595 

8S78 


2305 
0415 
8716 


2106 
0237 

8556 


1910 
0061 

8397 


1716 
9887 
8239 


1524 
9714 

8083 


1334 

9544 
7929 


1146 

9375 
7776 


0961 
9208 
7625 


32 64 96 
29 58 87 
26 52 78 


129 161 
115 144 
104 130 


20 


2-7475 


7326 


7179 


7034 


6889 


6746 


6605 


64G4 


6325 


6187 


24 47 71 


95 118 


21 
22 

23 


2-6051 
2-4751 
2-3559 


5916 
4627 
3445 


5782 
4504 
3332 


5649 
4383 
3220 


5517 
4262 
3109 


5386 
4142 
2998 


5257 
4023 

2889 


5129 
390G 

2781 


5002 
3789 
2673 


4876 
3673 
2566 


22 43 65 
20 40 60 
18 37 55 


87 108 
79 99 
74 92 


21 
25 
26 


2-24G0 
2-1445 
2-0503 


2355 
1348 
0413 


2251 
1251 
0323 


2148 
1155 
0233 


2045 
1060 
0145 


1943 
0965 
0057 


1842 
0872 
9970 


1742 

0778 
9883 


1642 

0686 
9797 


1543 
0594 
9711 


17 34 51 
16 31 47 
15 29 44 


68 85 
63 78 
58 73 


27 
28 
29 


1-9626 

1-8807 
1-8040 


9.542 
8728 
7966 


9458 
8650 
7893 


9375 
8572 
7820 


9292 
8495 

7747 


9210 

8418 
7675 


9128 
8341 
7603 


9047 
82G5 
7532 


89G7 
8190 
7461 


8887 
8115 
7391 


14 27 41 
13 26 38 
12 24 36 


55 68 
51 64 
48 60 


80 


1-7321 


7251 


7182 


7113 


7045 


6977 


6909 


6842 


6775 


6709 


11 23 34 


45 56 


81 

82 
88 


1-0643 

1-6003 

I 1-5399 


6577 
5941 
5340 


6512 
5880 
5282 


6447 
5818 
5224 


6383 
5757 
5166 


6319 
5697 
5108 


6255 
5637 
5051 


6191 
5577 
4994 


6128 
5517 
4938 


6066 
5458 
4882 


11 21 32 

10 20 30 
10 19 29 


43 53 
40 50 
38 48 


34 
86 
86 


; 1-4826 
1-4281 

1 1-3764 


4770 
4229 
3713 


4715 

417(5 
3663 


4659 
4124 


4605 
4071 
3564 


4550 
4019 
3514 


4496 
3968 
3465 


4442 
39 16 
3416 


43S8 
3865 
3367 


4335 
3814 
3319 


9 18 27 
9 17 26 
8 16 25 


36 45 
34 43 
33 41 


87 

3s 
30 


1-3270 
1 -23 19 


3222 
2753 

2305 


3175 
2708 

22(51 


3127 

2002 
2218 


3079 
2G17 
2174 


3032 
2572 
2131 


2985 
2527 
2088 


293S 
2482 
2045 


2892 
2437 
2002 


2846 
2393 

19(50 


8 16 23 
8 15 23 
7 14 22 


31 39 
30 38 
29 36 


16 


11918 


1875 


1833 


1792 


1750 


1708 


1667 


1626 


1585 


1544 


7 J4 21 


28 34 


41 
42 

43 


11504 
11106 

1-0724 


1463 
1067 

0686 


1423 
1028 

0(549 


1383 

0990 
OG12 


1343 
0051 

0575 


1303 
0913 


1263 
0S75 
0501 


1224 
0837 
0464 


1184 

0799 
0428 


1145 
07(51 
0392 


7 13 20 
6 13 19 

6 12 18 


26 33 
25 32 
25 31 


44 


1-0955 


0319 




0247 


0212 


0176 


0141 


0105 


0070 0035 


6 12 18 


24 30 



X.L. — Numbers in difference-columns to be subtracted, not added. 



NATURAL COTANGENTS. 



193 





I °' 


6' 


12' 


18' 


24' 


30' 


| 36' 


42' 


1 48' 


i 54' 


1 


2 3 4 5 


45 s 10 


0-9965 


0-9930 


0-9896 


0-9861 


0-9827 


0-9793 


0-9759 


0-9725 


0-9691 


6 


11 17 23 29 


46 
47 

4S 


•9G57 
•9325 
■9004 


9623 
9293 
8972 


9590 
9260 
8941 


9556 
9228 
8910 


9523 
9195 
8878 


9490 
9163 

1 8847 


9457 
9131 
8816 


9424 
9099 

8785 


9391 
9067 
8754 


9358 
9036 
8724 


6 11 17 
5 11 16 
5 10 16 


22 28 
21 27 
21 26 


49 
50 
51 


•8693 

■8391 


8662 
8361 
8069 


8632 
8332 
8040 


8601 
8302 

5012 


8571 

8273 
7983 


8541 
8243 

79-54 


8511 
8214 
7926 


S481 
8185 
7898 


8451 
8156 
7869 


8421 

8127 
7841 


5 
5 

5 


10 15 
10 15 
10 14 


20 25 
20 24 
19 24 


s 

53 
54 


•7513 
■7530 
■72&j 


7785 
7508 
7239 


7757 
74S1 
7212 


7729 
7454 
7186 


7701 
7427 
7159 


7673 
7400 
7133 


7646 
7373 

7107 


7618 
7346 
7080 


7590 
7319 
7054 


7563 

7292 
7028 


5 

5 
4 


9 14 

9 14 
9 13 


18 23 
18 23 
18 22 


55 '7002 


6976 


6950 


6924 


6899 


6573 


6847 


6822 


6796 


6771 


4 


9 13 


17 21 


56 -6745 

57 1-6494 

5S -6249 


6720 
6469 
6224 


6694 
6445 
6200 


6669 
6420 
6176 


6644 
6395 
6152 


6619 
6371 

6123 


6594 
6346 

6104 


6569 
6322 
6080 


6544 

6297 
6056 


6519 
6273 
6032 


4 

4 

4 


8 13 

8 12 
8 12 


17 21 
16 20 
16 20 


59 '6009 

60 H-5774 

61 -5543 


5985 
5750 
5520 


5961 
5727 
5498 


5938 
5704 
5475 


5914 
5681 
5452 


5890 
5658 
5430 


5867 

5635 


5844 
5612 
5384 


5820 

5589 
5362 


5797 
5566 
5340 


4 

4 
4 


8 12 
8 12 
8 11 


16 20 
15 19 
15 19 


62 
63 
64 


•5317 
•5095 

•4877 


5295 
5073 

4856 


5272 
5051 
48.34 


52.50 
5029 
4813 


5228 
5008 
4791 


5206 
4986 

4770 


5184 
4964 
4743 


5161 
4942 
4727 


5139 
4921 
4706 


5117 

4S99 
4684 


4 
4 
4 


7 11 
7 11 
7 11 


15 18 
15 18 

14 IS 


65 


•4663 


4642 


4621 


4599 


4578 


4557 


4536 


4515 


4494 


4473 


4 


7 10 ! 14 18 


66 
67 
6S 


•4452 
•4245 
•4040 


4431 
4224 
4020 


4411 
4204 
4000 


4390 
4183 
3979 


4369 
4163 
3959 


4:348 
4142 
3939 


4327 
4122 
3919 


4307 
4101 
3S99 


4286 

4081 
3879 


4265 
4061 
3859 


3 
3 

3 


7 10 14 17 
7 10 14 17 
7 10 13 17 


69 
70 
71 


•3839 
•3640 
•3443 


3819 
3620 
3424 


3799 
3600 
3404 


3779 
3581 
3385 


3759 
3561 
3365 


3739 
3541 
3346 


3719 
3522 
3327 


3699 
3502 
3307 


3679 
3482 
3288 


3659 
3463 
3269 


3 
3 

3 


7 10 
6 10 
6 10 


13 17 
13 17 
13 16 


72 
73 
74 


•3249 
•3057 

•2867 


3230 

3038 
2849 


3211 
3019 
2830 


3191 
3000 
2811 


3172 

2931 
2792 


3153 
2962 
2773 


31.34 
2943 
27.54 


3115 
2924 
2736 


3096 
2905 
2717 


3076 

2698 


3 

3 
3 


6 10 
6 9 

6 9 


13 16 
13 16 
13 16 


75 


•2679 


2661 


2642 


2623 


2605 


25S6 


2568 


2549 


2530 


2512 


3 


6 9 


12 16 


76 
77 
7S 


•2493 
•2309 
•2126 


2475 
2290 
2107 


2456 

2272 
2089 


2438 
22.54 
2071 


2419 
2235 
2053 


2401 
2217 
2035 • 


2382 
2199 
2016 


2364 
2180 
1998 


2345 
2162 
1980 


2327 

2144 
1962 


3 
3 
3 


6 9 
6 9 
6 9 


12 15 
12 15 

12 15 


79 
sO 

rij 


■1944 
•1763 
1584 


1926 
1745 
1566 


1908 
1727 
1548 


1890 
1709 
1530 


1871 
1691 

1512 


1853 
1673 
1495 


1835 
1655 
1477 


1817 
1638 
1459 


1799 
1620 
1441 


1781 
1602 
1423 


3 

3 
3 


6 9 
6 9 
6 9 


12 15 
12 15 
12 15 


S2 -1405 
83 -1228 
S4 -1051 


1388 
1210 
1033 


1370 
1192 
1016 


1352 
1175 
0998 


1334 
1157 
0981 


1317 
1139 
0963 


1299 
1122 
0945 


12S1 
1104 

0928 


1263 
10S6 
0910 


1246 
1069 
0892 


3 
3 
3 


6 9 

6 9 
6 9 


12 15 
12 15 
12 15 


85 i-0375 


0857 


0840 


0822 


0805 


0787 


0769 


0752 


0734 


0717 


3 


6 9 


12 15 


86 -0699 
S7 -0524 
SS -0.349 


0682 
0507 
0332 


0664 
0489 
0314 


0647 

0472 
0297 


0629 
0454 
0279 


0612 
0437 
0262 


0594 
0419 
0244 


0577 
0402 
0227 


0559 
0384 
0209 


0542 
0367 

0192 


3 
3 
3 


6 9 
6 9 
6 9 


12 15 
12 15 
12 15 


89 |-0175 


0157 


0140 


0122 


0105 


0087 


0070 


0052 


0035 


0017 


3 


6 9 


12 14 



X.B. — Xunibers in difference-columns to be subtracted, not added. 



INDEX. 



PAGE 

Amalgamation of zinc .... 67 

Archimedes' principle 27 

Balance, description of .... 18 

ratio of arms of 23 

rules for nse of 20 

sensibility of 22 

■weighing with 22 

Ballistic galvanometer 71 

Barometer, description of ... . 9 

corrections to reading of . . . 10 
tables of corrections for . . 154, 155 

Batteries, chromic acid 65 

LeClanche 65 

gravity 63 

storage 64 

water 144 

Boiling-point, determination of . . 

Boiling-points, table of 157 

Bottle, specific gravity 26 

Calculations with small quanti- 
ties 123 

Caliper, micrometer 11 

Candle-power 103 

Capacities, specific inductive, table 

of 171 

Cathetometer 16 

Capillary constant 44 

Capillary depression of mercury, 

table of 154 

Cell standard . 143 

Cells, galvanic, see Batteries. 

arrangement of 65 

Cements , 142 

Coincidences, time of vibration by . 31 

Collimation, line of 17 

Collimator of spectrometer . . . 108 

Conductivity, electric, specific . . 89 

thermal, table of . 161 



PAGE 

Connections, electrical ..... 67 

Commutator 66 

Computation of results 3 

with small quantities .... 123 
Constant of a tangent galvanom- 
eter 91 

Constants, computation of by least 

squares 124 

tables of, astronomical .... 176 

geodetical 175 

magnetic of U. S 172 

numerical 175 

physical 176 

Curvature of a lens 13, 104 

Curves, directions for plotting . . 3 



D'Arsonval galvanometer . . . 
Differential galvanometer ... 69, 
Dimensions of physical quantities . 

Dividing engine 

Density defined 

of a gas by Meyer's method . . 

of a liquid with bottle .... 

of a liquid by immersion . . . 

of a regular solid by dimensions 

of an irregular solid by immer- 
sion . . . . 

tables of, for air . 



gases . 
liquids 
solids . 
solutions 
water . 
Dew point, determination of 



148. 



70 
S9 
180 
15 
25 
28 
26 
28 
26 

28 
152 
151 
149 
149 
150 
151 

61 



Elasticity, modulus of, — 

by bending , . 

by stretching 

torsional ......... 

of solids, table of 163 



195 



196 



INDEX. 



Electrioity and Magnetism, intro- 
duction to 

Bleotrometer, quadrant 

Edelmann's . . . . 

Electromotive force, by comparison, 

with electrometer, 

of cells, table of . 

Electro-chemical equivalents, 

table Of , 

Errors of observation . . , 

discussion of ..... 
Expansion, thermal, coefficient of, 

tables 

Extension, measurement of . 
Fall of potential, resistance by- 
Fibres, quartz, — 

directions for drawing . . 

mounting . 

silvering 

soldering . 

Fraunhofer's lines 

tables of 

Freezing-mixtures table of 
Freezing-point of water . . 
Freezing-points, table of . . 
Friction, molecular; see Viscosity 
Fusion, heat of 

tables of 



Galvanometer, calibration of 
description of ballistic 

D'Arsonval 
differential 
mirror . 
pointer . 
tangent . 
Gauge, micrometer . . . 

vernier 

(g, directions for working 
Bilvering 
Gravity, acceleration of, with s 

pie pendulum 

Gravity, specific ; see Density 
Graphic representation . . . 



Heat, expansion coefficient of 

latent, of fusion .... 

of vaporization . . 

tables of . . • 

itic 

table of 



G2 

96 
97 
95 
9G 
1G9 

173 

1 

121 



160 

7-18 
84 

138 
139 
139 
140 
113 
167 
162 

157 

56 
158 

91 
71 
70 
69,89 
93 
67 
HI 
11 



132 
135 



54 
56 

152 
54 
159 



TAGE 

Hydrometer scales, tables for re- 
duction of 149 

Hygrometer, Darnell's 61 

Eygrometry 60 

table for 154 

Humidity 61 

Immersion, density by 28 

Index of refraction of a prism . . 110 

Indices of refraction, table of . . 169 
Inductive capacity, specific, table 

of 171 

Inertia, moments of, — 

formulas for 40 

determination of, with torsion 

pendulum 41 

Infinitesimals, calculations involv- 
ing 123 

Interpolation, graphic 6 

Latent heat, of fusion 54 

of vaporization . . 56 

tables of . . . 152 

Least squares, computation by . . 124 

Lenses, focal distance of ... . 105 

radius of curvature of, with 

spherometer 13 

radius of curvature with tele- 
scope 104 

Lever, optical 34 

Light, wave length of 115 

photometry 102 

polarizer 117 

Logarithms, use of tables .... 181 

tables 184 

Magnifying power of optical in- 
struments 107 

Magnetic constants for TJ. S. . . . 172 

inclination 72 

Magnetism, horizontal intensity of 

earth's field 73 

Manipulations with glass .... 132 

mercury . . . 144 

solder .... 140 

Mass, measurement of 18 

Melting-points, table of 157 

Mercury, capillary depression of, 

table 154 

directions for cleaning . ... 141 

surface tension of 46 



IXDEX. 



197 



PAGE 

Metric units, conversion to English, 177 

Micrometer gauge 11 

Micrometer, filar 14 

Middle elongations, time of vibra- 
tion by 41 

Mirror and scale 34, 62 

Modulus of elasticity, Young's, — 

by bending 38 

stretching 34 

torsional 39 

Moments of Inertia, — 40 

formulas for determination of . 41 

Newton's rings, colors of .... 168 

Notebook, suggestions concerning, 3 

Observations, errors of . . . 1, 123 

rejection of 2 

relative importance of ... . 2 

Optical lever 34, 62 

Oscillation, time of, — 

by coincidences 31 

by middle elongations .... 41 

reduction to small arc . . . 165 

Parallax, error of . 1 

Pendulum, simple, determination 

of g by 31 

correction for moment of ball . 33 

torsional, period of 38 

Photometer, Bunsen's 102 

Joly's 103 

Pitch, of a tuning fork, by siren . 101 

Prism, measurement of angle of . 109 

index of refraction of . . . . 110 

Potential, resistance by fall of . . 84 

Polarization of light, plane . . . 117 

circular . . 118 

Psychrometer, August's 60 

Quartz fibres, — 

directions for drawing .... 138 

mounting .... 139 

silvering 139 

soldering 140 

Refraction, index of 110 

indices of, tables of . 169 

Resistance-box, description of . . 71 
Resistance, electrical, of a battery. 

by alternating current .... 88 



PAGE 

Resistance, electrical, of a battery, 

by Mances' method 81 

Ohm's method S3 

of an electrolyte 87 

of a galvanometer by Thomson's 

method 80 

of a metallic conductor — 

by differential galvanometer, 89 

substitution 77 

Wheatstone's bridge . . 78 

specific, by fall of potential . . 84 

temperature, coefficient of . . S6 

table of specific 170 

Results, discussion of .... 3, 121 

Rotation of plane of polarization . 118 

Saccharimetry 118 

Shunts, law of 71 

Soldering, directions for .... 140 

quartz fibres 140 

Sound, pitch of, by siren .... 101 

tables 164 

velocity in air 100 

in solids .... 99 

tables 164 

Specific gravity, see Density ... 54 

heat 159 

inductive capacity .... 171 

Spectra, kinds of 112 

tables of 166 

Spectrometer, angle of prism with, 109 

Spectroscope 113 

Spectrum, diffraction 116 

solar 113 

analysis 112 

Spherometer, description of . . . 112 

thickness with ... 13 

curvature with ... 13 

Squares, least, discussion of . . . 124 

Standard cell, construction of . . 143 

Sugar, per cent of, in a solution . US 

Surface tension of water .... 44 

of mercury ... 46 

table of 162 

Siren, pitch of fork with .... 101 

Switch, reversing 66 

Tables 147 

barometer corrections .... 155 

capillarity 162 

capillary depression of mercury 541 

constants 175, 176 



108 



INDEX. 



Tables {continued)— page 

density 148-163 

dimensions of physical quanti- 
ties 180,181 

elasticity 183 

electricity 170, 171 

heat 157-162 

hygrometry 154 

light 166-169 

logarithms 184 

magnetism 172-174 

sound 1G4 

thermometer corrections . . . 15G 

units 177-179 

viscosity 1G3 

weight in vacuo 153 

Tangent galvanometer 91 

Telescope, line of collimation of . 17 
magnifying power of . . 107 

and scale 34, G2 

Tenths, estimation of 3 

Temperatures, critical 158 

Tension, surface, of water .... 44 

of mercury ... 46 

table of .... 162 

Thermometer, calibration of . . . 48 

table of corrections 

for 156 

wet and dry bulb . 61 
Time of periodic motion — 

by coincidences 31 

middle elongations 31 

Torsional elasticity 39 

Torsional pendulum 38 

Tuning-fork, pitch of with siren . 101 



PAGE 

Units, conversion of English to me- 

tric 178 

conversion of metric to English, 177 

Vapor density, determination of . 28 

Vaporization, heat of 56 

table 158 

Vernier, circular 109 

straight 7 

gauge 8 

Vibration number — 

of a fork 101 

of musical notes 164 

Vibration, period of — 

by coincidences 31 

by middle elongations .... 31 

reduction to small arc .... 165 

Viscosity of liquids 46 

table of 163 

Volume — 

of irregular solid 27 

a gas at different tempera- 
tures 153 

water at different temperatures, 151 

"Water battery, directions for mak- 
ing 144 

Wave length of light 116 

table 167 

Wheatstone's bridge 78 

box form 88 

Zinc, amalgamation of 67 



r 



46 SCIENCE. 



Primary Batteries 

By Professor HENRY S. Carhart, University of Michigan. Sixty- 
seven illustrations. 1 21110, cloth, 202 pages. Price, $1.50. 

THIS is the only book on this subject in English, except a 
translation. It is a thoroughly scientific and systematic 
account of the construction, operation, and theory of all the 
best batteries. An entire chapter is devoted to a description of 
standards of electromotive force for electrical measurements. 
An account of battery tests, with results expressed graphically, 
occupies forty pages of this book. 

The battery as a device for the transformation of energy is 
kept constantly in view from first to last ; and the final chapter 
on Thermal Relations concludes with the method of calculating 
electromotive force from thermal data. 

Professor John Trowbridge, Harvard University : I have found it of the 
greatest use, and it seems to me to supply a much needed want in the 
literature of the subject. 

Professor Eli W. Blake, Brown University : The book is very opportune, 
as putting on record, in clear and concise form, what is well worth know- 
ing, but not always easily gotten. 

Professor George F. Barker, University of Pennsylvania: I have read it 
with a great deal of interest, and congratulate you upon the admirable 
way in which you have put the facts concerning this subject. The latter 
portion of the book will be especially valuable for students, and I shall be 
glad to avail myself of it for that purpose. 

Professor John E. Davies, University of Wisconsin: I am so much pleased 
with it that I have asked all the electrical students to provide themselves 
with a copy of it. . . . I have assured them that if it is small in size, 
it is, nevertheless, very solid, and they will do well to study and work over 
it very carefully. . . . I find it invaluable. 

Albert L. Arner, Instructor in Physics, Iowa State University : I am using 
your work on Primary Batteries, and find it the best guide to practical 
results in my laboratory work of any thing that I have yet secured. It is 
a book we long have needed, and it is one of that kind which is not 
exhausted at a first reading. 

Professor Alex Macfarlane, University of Texas: Allow me to congratu- 
late you on producing a work which contains a great deal of information 
which cannot be obtained readily and compactly elsewhere. 






SCIENCE. 47 



Physics for University Students 

By Professor Henry S. Carhart. Part I. Mechanics, Sound, and 
Light, with 154 Illustrations. i2mo, cloth, 330 pages. Price, $1.50. 

THIS book, which is the outgrowth of long experience in 
teaching, offers a half-year's course in University Physics. 
It is strictly a text-book, and not a compendium or a cyclopaedia ; 
as it is intended to aid the teacher, and not to be a substitute for 
him, it leaves room for the personal equation in teaching. 

Particular attention has been given to the arrangement of 
topics, so as to secure a natural and logical sequence. In many 
demonstrations the method of the Calculus is used without its 
formal symbols ; and, in general, mathematics is called into ser- 
vice, not for its own sake, but wholly for the purpose of estab- 
lishing the relations of physical quantities. 

To simple harmonic motion, more attention is paid than 
is customary in elementary courses. Numerous references to 
other books are given in connection with the headings of the 
articles. 

It is believed that the book will be helpful to teachers who 
adopt the prevailing method of a combination of lectures and 
text-book instruction. 

Professor W. LeConte Stevens, Rensselaer Polytechnic Institute, Troy, 
N.Y.: After an examination of Carhart's University Physics, I have 
unhesitatingly decided to use it with my next class. The book is admira- 
bly arranged, clearly expressed, and bears the unmistakable mark of the 
work of a successful teacher. 

Professor Floriani Cajori, Colorado College: The strong features of his 
University Physics appear to me to be conciseness and accuracy of state- 
ment, the emphasis laid on the more important topics by the exclusion of 
minor details, the embodiment of recent researches whenever possible. 

Professor Charles Hill, State University, Seattle, Wash.: I believe the 
book is the very one we want. 

Professor A. A. Atkinson, Ohio University, Athens, O. : I am very much 
pleased with the book. The important principles of physics and the 
essentials of energy are so well set forth for the class of students for 
which the book is designed, that it at once commends itself to the 
teacher. I should be more than pleased to receive a copy of Part II. 
when it shall be published. 



48 SCIENCE. 



The Elements of Physics 

By Professor Henry S. Carhart, University of Michigan, and H. N. 
Chute, Ann Arbor High School. 121110, cloth, 392 pages. Price, #1.20. 

THIS is the freshest, clearest, and most practical manual on 
the subject. Facts have been presented before theories. 

The experiments are simple, requiring inexpensive apparatus, 
and are such as will be easily understood and remembered. 

Every experiment, definition, and statement is the result of 
practical experience in teaching classes of various grades. 

The illustrations are numerous, and for the most part new, 
many having been photographed from the actual apparatus set 
up for the purpose. 

Simple problems have been freely introduced, in the belief 
that in this way a pupil best grasps the application of a principle. 

The basis of the whole book is the introductory statement 
that physics is the science of matter and energy, and that noth- 
ing can be learned of the physical world save by observation and 
experience, or by mathematical deductions from data so obtained. 
The authors do not believe that immature students can profitably 
be set to rediscover the laws of Nature at the beginning of their 
study of physics, but that they must first have a clearly defined idea 
of what they are doing, an outfit of principles and data to guide 
them, and a good degree of skill in conducting an investigation. 

William H. Runyon, Armour Institute, Chicago : Carhart and Chute's text- 
book in Physics has been used in the Scientific Academy of Armour 
Institute during the past year, and will be retained next year. It has 
been found concise and scientific. We believe it to be the best book on 
the market for elementary work in the class-room. 

Professor M. A. Brannon, University of North Dakota, Grand Forks: I 
am glad to express the opinion, based on the use of this work in Elemen- 
tary Physics last year, at Fort Wayne, Ind., that it is the most logical and 
clear presentation of the subject with which I am acquainted. The prob- 
lems associated with the discussion of Physical phenomena, laws, and ex- 
periments serve the dual purpose of leading the scholar to reason, and 
put into practice the previous clearly and concisely stated principles of 
Elementary Physics. It is a book that will greatly elevate the standard 
of scholarship wherever used. 



SCIENCE. 49 



Professor H. N. Allen, University of Nebraska, Lincoln : Carhart and 
Chute's Physics is not used in the University, but it is recommended by 
us for use in high schools. I believe it to be one of the best books of 
its class published. 

Professor Arthur M. Goodspeed, University of Pennsylvania. I have not 
had time to read it in course, but have examined it in parts quite care- 
fully, and do not hesitate in saying that, for a book of its size and grade, 
I deem it the best one that has been brought to my attention. 

Professor John W. Johnson, University of Mississippi: I have examined 
Elements of Physics by Carhart and Chute, and I believe it combines the 
theoretical and practical in just the right proportion to make it a most 
efficient and valuable text-book. 

G. W. Krall, Manual Training School, St, Louis, Mo. : I have used Carhart 
and Chute's Physics during the past year with entire satisfaction to teacher 
and pupils. The book is fresh in presentation, omits the worn-out matter 
of ordinary text-books, and is clear and exact in statement. The problems 
are excellent and new ; and the book breathes the spirit of new methods. 
It is by far the best book published for secondary schools. We shall con- 
tinue its use the coming year. 

Dwight M. Miner, High School, Taunton, Mass.: After examining care- 
fully a number of books, I decided to adopt Carhart and Chute's Elements 
of Physics. After using it twenty-five weeks, I can say that it has fully 
come up to my expectations. I find the experiments well chosen, the 
explanations clearly put, and the arrangement logical. I am especially 
pleased with the chapter on Electricity. I shall continue to use the book. 

William F. Langworthy, Colgate Academy, Hamilton, NY. : After using 
Carhart and Chute's Physics for the past year, I can say that it is the very 
best text-book I have seen. It is almost perfectly adapted to our needs. 
We shall continue to use it. 

W. C. Peckham, Adelphi Academy, Brooklyn, N.Y.: Your Carhart and 
Chute's Physics on the whole impresses me as the best book out for a 
beginner to use in getting his first view of the general principles of the 
whole subject. 

C. F. Adams, High School, Detroit, Mich. : The Carhart and Chute's Phys- 
ics, which has been in use in my classes since last September, has given 
excellent satisfaction. The book is thoroughly scientific, and is abreast 
with modern thought and developments. Before using the book I was 
somewhat doubtful as to the ability of my pupils to grasp those subjects 
as presented in this book, but the results have been a pleasant surprise 
to me. 



SCIENCE. 



Principles of Chemical Philosophy 

By JOSIAH PARSONS COOKE, late Professor of Chemistry, Harvard Uni- 
versity. Revised Edition. Svo, cloth, 634 pages. Price, £3.50. 

THE object of this book is to present the philosophy of chem- 
istry in such a form that it can be made with profit the 
subject of college recitations. Part I. of the book contains a 
statement o( the general laws and theories of chemistry, together 
with so much of the principles of molecular physics as are con- 
stantly applied to chemical investigations. Part II. presents the 
scheme of the chemical elements, and is to be studied in con- 
nection with experimental lectures or laboratory work. 

Elements of Chemical Physics 

By Josiah Parsons Cooke. 8vo, cloth, 751 pages. Price, $4.50. 

THIS volume furnishes a full development of the principles 
of chemical phenomena. It has been prepared on a 
strictly inductive method ; and any student with an elementary 
knowledge of mathematics will be able easily to follow the 
course of reasoning. Each chapter is followed by a large num- 
ber of problems. 

Chemical Tables 

By Stephen P. Sharples. i2mo, cloth, 199 pages. Price, $2.00. 

P hysical Laboratory Practice 

By Professor A. M. Worthington, Royal Naval Engineering College, 
Davenport, Eng. With illustrations. i6mo, cloth, 316 pages. Price, jj5i.2o. 

IN preparing this manual, special effort has been made to select 
such experiments as shall give a good mental grip of the point 
to be illustrated, and to confine the attention to matters that can 
be fairly mastered by pupils of fourteen to sixteen years of age. 
The book contains : 
Mensuration, 23 Experiments. Heat, 42 Experiments. 

Hydrostatics, 15 Experiments. Magnetism, 55 Experiments. 
Mechanics, 39 Experiments'. Electricity, 73 Experiments. 
Elasticity, 20 Experiments. 



SCIENCE. 51 



Descriptive Inorganic General Chemistry 

A text-book for colleges, by Professor Paul C. Freer, University of 
Michigan. Revised Edition. Svo, cloth, 559 pages. Price, S3.00. 

IT aims to give a systematic course of Chemistry by stating 
certain initial principles, and connecting logically all the 
resultant phenomena. In this way the science of Chemistry 
appears, not as a series of disconnected facts, but as a harmo- 
nious and consistent whole. 

The relationship of members of the same family of elements 
is made conspicuous, and resemblances between the different 
families are pointed out. The connection between reactions is 
dwelt upon, and where possible they are referred to certain prin- 
ciples which result from the nature of the component elements. 

The frequent use of tables and of comparative summaries lessens 
the work of memorizing, and affords facilities for rapid reference 
to the usual constants, such as specific gravity, melting and boil- 
ing points, etc. These tables clearly show the relationship be- 
tween the various elements and compounds, as well as the data 
which are necessary to emphasize this relationship. They also 
exhibit the structural connection between existing compounds. 

Some descriptive portions of the work, which especially refer 
to technical subjects, have been revised by men who are actively 
engaged in those branches. In the Laboratory Appendix will 
be found a list of experiments, with descriptive matter, which 
materially aid in the comprehension of the text. 

Professor Walter S. Haines, Rush Medical College, Chicago.: The work is 
worthy of the highest praise. The typography is excellent, the arrange- 
ment of the subjects admirable, the explanations full and clear, and facts 
and theories are brought down to the latest date. All things considered, 
I regard it as the best work on inorganic chemistry for somewhat advanced 
general students of the science with which I am acquainted. 

Professor J. H. Long, Northwestern University. Evanston, III.: I have 
looked it over very carefully, as at first sight 7 was much pleased with both 
style and arrangement. Subsequent examination confirms the first opinion 
that we have here an excellent and a very useful text-book. It is a book 
which students can read with profit, as it is clear, systematic, and modern. 



MATHEMATICS. 

Principles of Plane Geometry 

By J. \V. MACDONALD, Agent of the Massachusetts Board of Educa- 
tion. l6mo, paper, 70 pages. Price, 30 cents. 

THIS book may be described as an excellent Geometry with- 
out the demonstrations. Even the axioms and the defini- 
tions are given as questions. The pupil is expected to do a 
great deal of thinking, and not much memorizing. 

Public Opinion, Washington, D.C.: It is time that teachers should see that 
what they gain in greater ground covered by the old method, they lose in 
power of mind on the part of pupils. Why is it that so many are going 
out into the world as " scientific " workers ? Because by the methods of 
training, students in science are not obliged to make a fatal leap when they 
go from training-schools into actual work. Mathematics would become 
just as fascinating if the same methods were employed. The elementary 
methods of teaching geometry must be improved in this direction before 
the best results are reached. Mr. MacDonald speaks from a full expe- 
rience and a demonstration in his own class-room of the methods here set 
forth. They are attracting wide attention. We know .they will win their 
way with scores of aspiring teachers. 

A Primary Algebra 

By J. W. MacDonald, Agent of the Massachusetts Board of Educa- 
tion. i6mo, cloth. The Complete Edition, 214 pages (containing the 
Teacher's Guide, the Student's Manual and Answers to Problems). 
Price, 75 cents. The Student's Manual, cloth, SS pages. Price, 30 cents. 

THE purpose of this book is made evident by its title. For 
grammar-school use it covers the work of about one year. 
In a high school or academy it can be finished in less time. 

It is published in two parts : TJie TeacJiers Guide and The 
Student's Manual. The former contains theory, explanations, 
definitions, etc., and can be used as circumstances require. 
The latter, lesson by lesson, furnishes examples for class-drill 
and for the student's home work. The Manual is published 
separately, and is the only part which the pupils will need. 

This arrangement enables the teacher to arouse the interest of 
the pupils and stimulate inquiry, to develop the principles in 
logical order, and allow the deduction of the definitions and 
rules from actual practice. 



MATHEMATICS. 53 



A College Algebra 



By Professor J. M. Taylor, Colgate University, Hamilton, N.Y. 
i6mo, cloth, 326 pages. Price, $1.50. 

A VIGOROUS and scientific method characterizes this book. 
In it equations and systems of equations are treated as 
such, and not as equalities simply. 

A strong feature is the clearness and conciseness in the state- 
ment and proof of general principles, which are always followed 
by illustrative examples. Only a few examples are contained in 
the First Part, which is designed for reference or review. The 
Second Part contains numerous and well selected examples. 

Differentiation, and the subjects usually treated in university 
algebras, are brought within such limits that they can be success- 
fully pursued in the time allowed in classical courses. 

Each chapter is as nearly as possibly complete in itself, so 
that the order of their succession can be varied at the discretion 
of the teachers. 

Professor W. P. Durfee, Hob art College, Geneva, N.Y. : It seems to me a 
logical and modern treatment of the subject. I have no hesitation in pro- 
nouncing it, in my judgment, the best text-book on algebra published in 
this country. 

Professor Geo. C. Edwards, University of California. ; It certainly is a 
most excellent book, and is to be commended for its consistent conciseness 
and clearness, together with the excellent quality of the mechanical work 
and material used. 

Professor Thos. E. Boyce, Middlebury College, Vt.: I have examined with 
considerable care and interest Taylor's College Algebra, and can say that 
I am much pleased with it. I like the author's concise presentation of 
the subject, and the compact form of the work. 

Professor H. M. Perkins, Ohio Wesley a?i University.: I think it is an 
excellent work, both as to the selection of subjects, and the clear and 
concise method of treatment. 

Professor S. J. Brown, University of Wisconsin : I am free to say that 
it is an ideal work for elementary college classes. I like particularly the 
introduction into pure algebra, elementary problems in Calculus, and ana- 
lytical growth. Of course, no book can replace the clear-sighted teacher ; 
for him, however, it is full of suggestion. 



54 MATHEMATICS. 



An Academic Algebra 

By Professor J. M. TAYLOR, Colgate University, Hamilton, N.Y. i6mo, 
cloth, 348 pages. Price, ^1.00. 

THIS book is adapted to beginners of any age, and covers 
sufficient ground for admission to any American college or 
university. In it the fundamental laws of number, the literal 
notation, and the method of solving and using the simpler 
forms of equations, are made familiar before the idea of alge- 
braic number is introduced. The theory of equivalent equa- 
tions and systems of equations is fully and clearly presented. 
Factoring is made fundamental in the study and solution of 
equations. Fractions, ratios, and exponents are concisely and 
scientifically treated, and the theory of limits is briefly and 
clearly presented. 

Professor C. H. Judson, Furman University, Greenville, S.C.: I take great 
pleasure in acknowledging the receipt of Taylor's Academic Algebra. I 
regard this and his college treatise as among the very best books on the 
subject, and shall take pleasure in commending the Academic Algebra to 
the schools of this State. 

Professor E. P. Thompson, Miami University, Oxford, O. : I find the claims 
made in your notice of publication well sustained, and that the book is 
compact, well printed, presenting just the subjects needed in preparation 
for college, and in just about the right proportion, and simply presented. 
I like the treatment of the theory of limits, and think the student should 
be introduced early to it. I am more pleased with the book the more 
I examine it. 

W. A. Ingalls, Principal, Marathon, N.Y.: I have waited some time before 
acknowledging the receipt of Taylor's Academic Algebra, in order to speak 
more understandingly of its merits. After measuring it by means of 
others which I have used in the class-room, I think it admirably suited for 
our regents' schools. More than this, it contains much that is valuable 
that is within the comprehension of the average student which is not found 
in other books of like grade. It is a scholarly book. 

Charles Henry Douglas, Principal of High School, Hartford, Conn. : The 
book is a good one, and contains more algebra than any other book of its 
size on the market. It is shorn of much of the " padding " that creeps into 
text-books on every subject, and deals with the essentials in a clear, vigor- 
;md progressive way. The treatment of factoring, and the emphasis 
put upon the importance of the equation, are particularly excellent. 



MATHEMATICS. 55 



Professor Geo. A. Harter, Delaware College, Newark, Del. : I have read it 
with much pleasure. Its whole plan and execution are so good I shall not 
attempt to particularize in my praise. I shall recommend it unhesitat- 
ingly whenever I have an opportunity. The typography and mechanical 
make-up are in keeping with the excellent contents. Indeed, author and 
publishers have produced a little gem of a text-book, and I am not sure 
but the publishers .have contributed as much to the attractiveness and value 
of the book as Dr. Taylor by putting it in such a model dress. 

L. P. Jocelyn, High School, Ann Arbor, Mich. : I examined it quite thor- 
oughly, and like it better than any of the many books I have examined. 

Professor W. B. Smith, Tulane University, New Orleans, La.: The gen-, 
eral air of this Algebra is very business-like. The author wastes few words 
in preliminaries, but closes quickly and earnestly with matters as they 
come to hand. The problems are exceedingly numerous and apparently 
well-chosen ; and on the whole the book would seem to be eminently teach- 
able. The author has strengthened the common presentation by calling 
particular attention to the doctrine of equivalent systems of equations, and 
has briefly sketched the Theory of Limits, making one extremely impor- 
tant application of it to the doctrine of Incommensurables, for which he 
will receive thanks from teachers of algebra. . . . 

Professor Taylor's book is a hopeful sign of the times, and teachers that use 
it — of whom may there be many — will almost certainly be pleased. 

Robert M. King, High School, Indianapolis, Ind. : I am much pleased with 
it. It is clear, concise, free from unnecessary material, and its treatment 
of factoring quadratics, and of other subjects, is very fine. 

E. P. Sisson, Colgate Academy, Hamilton, N.Y. : The book is conspicuously 
meritorious : first, in the clear distinction made between arithmetical and 
algebraic number, which lies at the foundation of a clear and comprehen- 
sive understanding of the science of algebra ; second, the introduction at 
the very first of the equation as an instrument of mathematical investiga- 
tion; third, . . . Dr. Taylor's presentation of the doctrine of equivalency 
is clear and rigid ; . . . fourth, the treatment of the subject of factoring 
is concise, comprehensive, and logical. ... I am using the book with 
satisfaction in my own classes. 

T. F. Kane, Superintendent of Schools, Naugatuck, Conn. : The book bears 
evidence of being prepared by a careful teacher. The fact that algebra is 
the science of the equation is emphasized throughout. Xo other text-book 
on the subject with which I am acquainted meets my ideal as nearly as this 
one does. The book is all the author claims. 

George A, Knapp, Professor of Mathematics, Olivet College, Olivet, Mich. : 
Professor Taylor's Algebra is clear and simple, yet thoroughly mathemati- 
cal. It is well adapted to the purpose that elementary algebra subserves, — 
laying the foundations for mathematical ability. 



50 SCIENCE. 



Anatomy, Physiology, and Hygiene 

A Manual for the Use of Colleges, Schools, and General Readers. By 
Jerome Walker, m.d. 121110, cloth, 427 pages. Price, $1.20. 

THIS book was prepared with special reference to the require- 
ments of high and normal schools, academies, and colleges, 
and is believed to be a fair exponent of the present condition of 
the science. Throughout its pages lessons of moderation are 
taught in connection with the use of each part of the body. 
The subjects of food, and of the relations of the skin to the 
various parts of the body and to health, are more thoroughly 
treated than is ordinarily the case. All the important facts are 
so fully explained, illustrated, and logically connected, that they 
can be easily understood and remembered. Dry statements are 
avoided, and the mind is not overloaded with a mass of technical 
material of little value to the ordinary student. 

The size of type and the color of paper have been adopted in 
accordance with the advice of Dr. C. R. Agnew, the well-known 
oculist. Other eminent specialists have carefully reviewed the 
chapters on the Nervous System, Sight, Hearing, the Voice, and 
Emergencies, so that it may justly be claimed that these impor- 
tant subjects are more adequately treated than in any other school 
Physiology. 

The treatment of the subject of alcohol and narcotics is in 
conformity with the views of the leading physicians and physiol- 
ogists of to-day. 

The Nation, New York : Dr. Jerome Walker's Anatomy, Physiology, and 
Hygiene appears an almost faultless treatise for colleges, schools, and 
general readers. Careful study has not revealed a serious blemish ; its 
tone is good, its style is pleasant, and its statements are unimpeachable. 
We cordially commend it as a trustworthy book to all seeking information 
about the body, and how to preserve its integrity. 

Journal of the American Medical Association : For the purposes for 
which it is written, it is the most interesting and fairest exponent of 
present physiological and hygienic knowledge that has ever appeared. 
It should be used in every school, and should be a member of every family, 
— more especially of those in which there are young people. It is a 
pleasure to read and review such an excellent book. 



SCIENCE. 57 



Professor J. C. Richardson, M.D., Late of the University of Pennsylvania: 
I cordially congratulate you upon the clear, accurate, and attractive way in 
which you have set forth the great facts of our human anatomy and physi- 
ology, and founded upon them the laws of hygiene. I hope and believe 
your excellent work will do much to instruct the rising generation in the 
priceless knowledge of how to preserve health, and attain long life for 
themselves and for their children after them. 

Charles S. Moore, Principal of High School, New Bedford, Mass. : I can 
speak in terms of the highest commendation of Walker's Physiology as a 
text-book. An experience of two years with it enables me to say that I 
consider it the best text-book on physiology that is published. 

John W. Wyatt, Principal of High School, Lynchburg, Va. : The work is 
indeed a treatise of rare excellence. 

Jas. A. Merrill, State Normal School, Warrensburg, Mo. : The books are 
entirely satisfactory, and are all that could be desired. 

E. H. Russell, Principal of State Normal School, Worcester, Mass.: 
Walker's Physiology is used in our classes more than any other text-book 
on that subject, and gives good satisfaction to teachers and students. It 
is clear, comprehensive, and conveniently arranged for practical use. 

Geo. H. Tracy, Principal of High School, Water bziry, Conn.: As to 
Walker, I am glad to say that I know the book, and have tested its merit 
by work in the class. I consider it decidedly the best work I know of 
for high schools and academies. 

W. K. Hill, Carthage, III.: It was a positive pleasure to me to examine 
Walker's Physiology, as it always is to find something good and progres- 
sive in scientific text-books. 

W. J. Wolverton, Lock Haven, Pa.: It is the best work on physiology and 
Hygiene published so far as I have seen. 

Warren Craig, Principal of High School, Warren, 0. : It is a fascinating 
book for the general reader, and at the same time, by its accuracy, com- 
mends itself to the scientific man. In it there is no need of the notes and 
explanations used to complete or illustrate similar books. But the work 
deserves higher praise than to be compared with many physiologies used in 
our high schools and academies. Its language is plain, clear, and always 
intelligible. The author has secured the two requisites of an ideal text- 
book, — breadth of scope without being superficial, and conciseness of ex- 
planation without being technical. 

0. W. Collins, M.D., Superintendent of Schools, Framingham, Mass.: 
We have used Walker's Physiology for the past four years. Every one 
acquainted with the science, after giving this book a thorough trial, will 
admit it to be the very best published for high school use. 



58 SCIENCE. 



Herbarium and Plant Descriptions 

Designed by Professor Edward T. Nelson, Ohio Wesleyan University. 
Portfolio, 7^ X 10 inches. Price, 75 cents. Adapted to any Botany. 

THIS is an herbarium and plant record combined, enabling 
the student to preserve the specimens together with a 
record of their characteristics. 

A sheet of four pages is devoted to each specimen. The first 
page contains a blank form, with ample space for a full descrip- 
tion of the plant, and for notes of the circumstances under 
which it was collected. The pressed specimen is to be mounted 
on the third page, and the entire sheet then serves as a species- 
cover. Each portfolio contains fifty sheets, which are separate, 
so as to permit of scientific rearrangement after mounting the 
specimens. 

The preliminary matter gives full directions for collecting, 
pressing, and mounting plants, as well as a synopsis of botani- 
cal terms. 

The portfolio is strong, durable, and attractive in appearance. 

In the class-room and in the field this work has been found 
helpful and stimulating. It encourages observation and research, 
and leads to an exact knowledge of classification. 

Professor D. P. Penhallow, McGill University, Montreal, Can. : The idea 
is a good one, and well carried out. I am sure it will prove most useful in 
the botanical work of schools and academies, for which I would strongly 
recommend it. 

Professor G. H. Perkins, University of Vermont, Burlington, Vt. : It is the 
best thing of the sort I have seen ; very attractive and very helpful to 
beginners in calling attention to points that would be overlooked. 

Professor B. P. Colton, Normal University, III. : It is a very ingenious ar- 
rangement, and neatly gotten up. It speaks well for the publishers, as 
well as the designer. It is the neatest scheme of the kind I have seen. 

0. D. Robinson, Principal of High School, Albany, N.Y.: It appears to me 
to be a very complete arrangement, admirable in every respect, and very 
moderate in price. 

F. 3. Hotaling, Formerly Principal of High School, Framingham, Mass.: 
Last year's work in botany was made so much more interesting and valua- 
ble by the use of the Herbarium that we find it now a necessity. 



MISCELLANEOUS 59 



Ancient Greece 

From the earliest times down to 146 B.C. By Robert F. Pennell, 
Principal of State Normal School, Chico, Cal. Revised Edition, with 
Plans and Colored Maps. i6mo, cloth, 193 pages. Price, 60 cents. 



Ancient Rome 



From the earliest times down to 476 a.d* By Robert F. Pennell. 
Revised Edition, with Plans and Colored Maps. i6mo, cloth, 284 pages. 
Price, 60 cents. 

IN these books the leading facts are presented in a concise and 
readable form. Minor details and unimportant names are 
omitted. The maps and plans have been drawn and engraved 
especially for the books, and contain ail the data, and only the 
data, necessary for following the story. 

The Index serves also as a key to the pronunciation of proper 
names. Examination papers used at Harvard, Yale, and by the 
Regents of the University of New York, are added in an appendix. 

Most teachers prefer a brief manual for a text-book in the 
hands of the pupils. It is easy to assign outside reading for 
special subjects in which it may be desired to spend extra time ; 
and the teacher can use his own judgment in selecting the topics 
to fit the pupils and the time at his disposal. Pennell's histories 
contain enough matter, if carefully studied, to enable a student 
to pass the entrance examinations at any American college or 
university. 

W. McD. Halsey, Collegiate School, 34 West Fortieth Street, New York City : 
PennelFs Greece and Rome are well adapted to their purpose. Having 
used the Greece last year, I will now add the Rome to the list of my 
text-books. The clear statements of the author, and the fine typographical 
work, commend the books at sight ; and the enthusiasm of both teachers 
and pupils shows that Mr. Pennell has made books suited to our need. 

H. B. Knox, Friends' School, Providence, R.I.: You are doing us all a ser- 
vice by putting the history of these nations into such an ideal form. 

Burr Lewis, Lincoln, Neb. : The additions only make a more excellent 
compound of Roman History for use in the class-room. It was excellent 
before. 



60 MISCELLANEOUS. 



Hamilton's Metaphysics 

Collected, Arranged, and Abridged by Francis Bo wen, late of Harvard 
University. i2mo, cloth, 571 pages. Price, #1.50. 

A Treatise on Logic 

Comprising both the Aristotelian and Hamiltonian Analyses of Logical 
Forms, and some chapters of Applied Logic, By Francis Bo wen. 
i2mo, cloth, 465 pages. Price, #1.25. 

Constitution of the United States 

With Brief Comments ; and Incidental Comments on the Constitutions 
of England and France. By J. T. Champlin, late President of Colby 
University. i6mo, cloth, 205 pages. Price, 80 cents. 

Democracy in America 

By Alexis De Tocqueville. Translated by Henry Reeve. 
Edited, with Notes, by Francis Bowen. In 2 vols. 8vo, cloth, 
Price, $4.00. 

American Institutions 

By Alexis De Tocqueville. Translated by Henry Reeve. Re- 
vised and edited, with Notes, by Francis Bowen. i2mo, cloth, 582 
pages. Price, $1.20. 

THIS publication is identical with Volume I. of the Democ- 
racy in America. It is issued in its present style to 
furnish the most valuable portion of the work in a cheaper 
form, and with especial reference to its use as a text-book. 

The Academy Class Register 

Price, delivered in any part of the United States, $1.50 per dozen. 
HIS is the cheapest Class Register published. It is ruled 



T 



for five recitations a week for twenty weeks, with space for 
summary each week, and for average each month. It will hold 
the names of eleven classes of twenty-seven pupils each. Names 
need be written only once during a term of twenty weeks. The 
paper is so finished that either ink or pencil may be used. 



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